FLORIDA

10

Sequences, Series, and the Binomial Theorem

MA.912.D.11.2

10.1 Define and Use Sequences and Series

MA.912.D.11.1

10.2 Analyze Arithmetic Sequences and Series

MA.912.D.11.1

10.3 Analyze Geometric Sequences and Series

MA.912.D.11.4

10.4 Find Sums of Infinite Geometric Series

Prep. MA.912.A.4.12 MA.912.A.4.12

10.5 Apply the Counting Principle and Permutations 10.6 Use Combinations and the Binomial Theorem

Before Previously, you learned the following skills, which you’ll use in Chapter 10: solving equations, solving systems of equations, simplifying expressions, and multiplying binomials.

Prerequisite Skills VOCABULARY CHECK Copy and complete the statement. 1. The coefficient of x 2 in the expression 3x 3 2 15x 2 1 4 is ? . 2. The expressions x 1 3 and 2x 2 1 are examples of binomials because they

have ? terms.

SKILLS CHECK Solve the equation. Check your solution. (Prerequisite skill for 10.2) 3. 10 2 3x 5 28

4. 11x 1 9 5 3x 1 17

5. 2x 1 3 5 26 2 x

Solve the system using any algebraic method. (Prerequisite skill for 10.3) 6. 3x 1 y 5 0

22x 2 4y 5 230

7. 2x 2 2y 5 10

x 1 y 5 210

8. 4x 2 5y 5 25

0.5x 1 1.5y 5 18.5

Simplify the expression. (Prerequisite skill for 10.5) 6p5p4p3 9. }}}}} 2p1

13 p 12 p 11 10. }}}}} 10 p 9 p 8

8p7p6p5p4 11. }}}}}}} 5p4p3p2p1

Find the product. (Prerequisite skill for 10.6) 12. (x 1 y)3

13. (5x 1 1)3

14. (3x 2 2y)3

For prerequisite skills practice, go to thinkcentral.com

Take-Home Tutor for problem solving help at www.publisher.com

676

Now In Chapter 10, you will apply the key ideas listed below and reviewed in the Chapter Summary on page 729. You will also use the key vocabulary listed below.

Key Ideas 1 Analyze sequences 2 Find sums of series 3 Use permutations and combinations KEY VOCABULARY • sequence, p. 678

• arithmetic sequence, p. 686

• geometric series, p. 697

• terms of a sequence, p. 678

• common difference, p. 686

• partial sum, p. 703

• series, p. 680

• arithmetic series, p. 688

• permutation, p. 713

• summation notation, p. 680

• geometric sequence, p. 695

• combination, p. 719

• sigma notation, p. 680

• common ratio, p. 695

• binomial theorem, p. 722

Why? You can use the fundamental counting principle and permutations to calculate the number of choices for a situation. For example, you can count the number of possible outcomes of an event or the number of ways to complete a task.

Algebra The animation illustrated below for Exercise 69 on page 718 helps you answer this question: How does the number of clothing choices affect the number of different ways you can dress mannequins in a display?

4/0

4 3HIRT

0OLO

,ONG3LEEVE

"/44/-



3HORTS

*EANS

4OTALNUMBEROFDISPLAY  CHOICESFORFIRSTMANNEQUIN 3TART

4OTALNUMBEROFDISPLAYCHOICES  FORSECONDMANNEQUIN

4OTALNUMBEROFDISPLAYCHOICESFORBOTHMANNEQUINS  #HECK!NSWER

Different outfits for a store display can be made using several tops and bottoms.

Find the total number of possible displays if there are one, two, or more mannequins.

"MHFCSB Go to thinkcentral.com

Other animations for Chapter 10: pages 689, 696, and 703

677

FLORIDA

Define and Use Sequences and Series

10.1

You used function notation to describe relationships.

Before

MA.912.D.11.2 Use sigma notation to describe series. Preparation for MA.912.D.11.1 Define arithmetic and geometric sequences and series.

Now

So you can find angle measures in a skylight, as in Ex. 63.

Why?

Key Vocabulary • sequence • terms of a sequence • series • summation notation • sigma notation

For Your Notebook

KEY CONCEPT Sequences

A sequence is a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. The values in the range are called the terms of the sequence. Domain: 1 Range:

2

3

4 ... n

a1 a2 a3 a 4 . . . an

The relative position of each term Terms of the sequence

A finite sequence has a limited number of terms. An infinite sequence continues without stopping. Finite sequence:

Infinite sequence:

2, 4, 6, 8

2, 4, 6, 8, . . .

A sequence can be specified by an equation, or rule. For example, both sequences above can be described by the rule an 5 2n or f(n) 5 2n.

EXAMPLE 1

Write terms of sequences

Write the first six terms of (a) an 5 2n 1 5 and (b) f (n) 5 (23) n 2 1. Solution a. a1 5 2(1) 1 5 5 7

b. f(1) 5 (23)1 2 1 5 1 221

a2 5 2(2) 1 5 5 9

2nd term

f(2) 5 (23)

a3 5 2(3) 1 5 5 11

3rd term

f (3) 5 (23) 3 2 1 5 9

a4 5 2(4) 1 5 5 13

✓

1st term

4th term

f(4) 5 (23)

5 23

2nd term 3rd term

421

5 227

4th term

521

5 81

5th term

a5 5 2(5) 1 5 5 15

5th term

f(5) 5 (23)

a6 5 2(6) 1 5 5 17

6th term

f(6) 5 (23) 6 2 1 5 2243

GUIDED PRACTICE

1st term

6th term

for Example 1

Write the first six terms of the sequence. 1. an 5 n 1 4

678

2. f(n) 5 (22) n 2 1

Chapter 10 Sequences, Series, and the Binomial Theorem

n 3. an 5 } n11

WRITING RULES If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the nth term of the sequence.

EXAMPLE 2 WRITE RULES If you are given only the first several terms of a sequence, there is no single rule for the nth term. For instance, the sequence 2, 4, 8, . . . can be given by an 5 2n or

an 5 n2 2 n 1 2.

Write rules for sequences

Describe the pattern, write the next term, and write a rule for the nth term of the sequence (a) 21, 28, 227, 264, . . . and (b) 0, 2, 6, 12, . . . . Solution a. You can write the terms as (21) 3, (22) 3, (23) 3, (24) 3, . . . . The next term is

a5 5 (25) 3 5 2125. A rule for the nth term is an 5 (2n) 3. b. You can write the terms as 0(1), 1(2), 2(3), 3(4), . . . . The next term is

f(5) 5 4(5) 5 20. A rule for the nth term is f (n) 5 (n 2 1)n. GRAPHING SEQUENCES To graph a sequence, let the horizontal axis represent the position numbers (the domain) and the vertical axis represent the terms (the range).

EXAMPLE 3

Solve a multi-step problem

RETAIL DISPLAYS You work in a grocery store

and are stacking apples in the shape of a square pyramid with 7 layers. Write a rule for the number of apples in each layer. Then graph the sequence. Solution Make a table showing the number of fruit in the first three layers. Let an represent the number of apples in layer n. Layer, n Number of apples, an

1

2

3

1 5 12

4 5 22

9 5 32

STEP 2 Write a rule for the number of apples in AVOID ERRORS Although the plotted points in Example 3 follow a curve, do not draw the curve because the sequence is defined only for integer values of n.

✓

each layer. From the table, you can see that an 5 n2.

STEP 3 Plot the points (1, 1), (2, 4), (3, 9), . . . , (7, 49). The graph is shown at the right.

GUIDED PRACTICE

Number of apples

STEP 1

an 48 32 16 0

0

2

4 6 Layer

n

for Examples 2 and 3

4. For the sequence 3, 8, 15, 24, . . . , describe the pattern, write the next term,

graph the first five terms, and write a rule for the nth term. 5. WHAT IF? In Example 3, suppose there are 9 layers of apples. How many

apples are in the 9th layer?

10.1 Define and Use Sequences and Series

679

For Your Notebook

KEY CONCEPT Series and Summation Notation

When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite. Finite series:

2141618

Infinite series:

21416181...

You can use summation notation to write a series. For example, the two series above can be written in summation notation as follows: 4

READING When written in summation notation, this series is read as “the sum of 2i for values of i from 1 to 4.”

∑ 2i

21416185

`

21416181...5

i51

∑ 2i

i51

For both series, the index of summation is i and the lower limit of summation is 1. The upper limit of summation is 4 for the finite series and ` (infinity) for the infinite series. Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written S.

EXAMPLE 4

Write series using summation notation

Write the series using summation notation. a. 25 1 50 1 75 1 . . . 1 250

2 1 3 1 4 1... 1 1} b. } } } 4 5 3 2

Solution a. Notice that the first term is 25(1), the second is 25(2), the third is 25(3),

and the last is 25(10). So, the terms of the series can be written as: ai 5 25i where i 5 1, 2, 3, . . . , 10 The lower limit of summation is 1 and the upper limit of summation

is 10.

10

c The summation notation for the series is

∑ 25i.

i51

b. Notice that for each term the denominator of the fraction is 1 more

than the numerator. So, the terms of the series can be written as: i where i 5 1, 2, 3, 4, . . . ai 5 } i11

The lower limit of summation is 1 and the upper limit of summation

is infinity.

`

c The summation notation for the series is

i ∑} . i11

i51

✓

GUIDED PRACTICE

for Example 4

Write the series using summation notation.

680

6. 5 1 10 1 15 1 . . . 1 100

4 1 9 1 16 1 . . . 1 1} 7. } } } 5 10 17 2

8. 6 1 36 1 216 1 1296 1 . . .

9. 5 1 6 1 7 1 . . . 1 12

Chapter 10 Sequences, Series, and the Binomial Theorem

INDEX OF SUMMATION The index of summation for a series does not have to

be i—any letter can be used. Also, the index does not have to begin at 1. For instance, the index begins at 4 in the next example.

EXAMPLE 5 AVOID ERRORS Be sure to use the correct lower and upper limits of summation when finding the sum of a series.

Find the sum of a series

Find the sum of the series. 8

∑ (3 1 k2) 5 (3 1 42) 1 (3 1 52) 1 (3 1 62) 1 (3 1 72) 1 (3 1 82)

k54

5 19 1 28 1 39 1 52 1 67 5 205

SPECIAL FORMULAS For series with many terms, finding the sum by adding the

terms can be tedious. Below are formulas you can use to find the sums of three special types of series.

For Your Notebook

KEY CONCEPT Formulas for Special Series Sum of n terms of 1

Sum of first n positive integers

n

n

∑15n

n

n(n 1 1) ∑i5 } 2

i51

EXAMPLE 6

Sum of squares of first n positive integers n(n 1 1)(2n 1 1) ∑ i2 5 } 6

i51

i51

Use a formula for a sum

RETAIL DISPLAYS How many apples are in the stack in Example 3 on page 679?

Solution From Example 3 you know that the ith term of the series is given by ai 5 i2 where i 5 1, 2, 3, . . . , 7. Using summation notation and the third formula listed above, you can find the total number of apples as follows: 7

12 1 2 2 1 . . . 1 7 2 5

1 1)(2 p 7 1 1) 7(8)(15) ∑ i2 5 7(7 } 5 } 5 140 6 6

i51

c There are 140 apples in the stack. Check this by actually adding the number of apples in each of the seven layers.

✓

GUIDED PRACTICE

for Examples 5 and 6

Find the sum of the series. 5

10.

∑ 8i

i51

7

11.

∑ (k 2 2 1)

k53

34

12.

∑1

6

13.

i51

∑n n51

14. WHAT IF? Suppose there are 9 layers in the apple stack in Example 3. How

many apples are in the stack? Explain using summation notation. 10.1 Define and Use Sequences and Series

681

10.1

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS18 for Exs. 19, 47, and 65

★

MA.912.D.11.2, Preparation for MA.912.D.11.1

5 STANDARDIZED TEST PRACTICE Exs. 2, 57, 58, 64, and 67

SKILL PRACTICE 1. VOCABULARY Copy and complete: Summation notation is also called

? .

2. ★ WRITING Explain the difference between a sequence and a series. EXAMPLE 1 on p. 678 for Exs. 3–14

WRITING TERMS Write the first six terms of the sequence.

3. an 5 n 1 2

4. an 5 6 2 n

5. an 5 n2

7. an 5 4n 2 1

8. an 5 2n2

9. f(n) 5 n2 2 5

4 11. f(n) 5 2} n EXAMPLE 2 on p. 679 for Exs. 15–26

3 12. an 5 } n

6. f(n) 5 n3 1 2

2n 13. an 5 } n12

10. an 5 (n 1 3)2 n 14. f(n) 5 } 2n 2 1

WRITING RULES For the sequence, describe the pattern, write the next term, and write a rule for the nth term.

15. 1, 6, 11, 16, . . .

16. 1, 2, 4, 8, . . .

17. 24, 8, 212, 16, . . .

18. 2, 9, 28, 65, . . .

2, 2, 2 , . . . 2, } 19. } } } 3 6 9 12

2, 4, 6, 8, . . . 20. } } } } 3 4 5 6

2, 3, 4, 5, . . . 1, } 21. } } } } 4 4 4 4 4

3 , 5 , 7 ,... 1 ,} 22. } } } 10 20 30 40

23. 3.1, 3.8, 4.5, 5.2, . . .

24. 4.2, 2.6, 1, 20.6, 22.2, . . .

25. 1.2, 4.2, 9.2, 16.2, . . .

26. 9, 16.8, 24.6, 32.4, . . .

EXAMPLE 3

GRAPHING SEQUENCES Graph the sequence.

on p. 679 for Exs. 27–35

27. 22, 25, 28, 211, 214

28. 2, 4, 8, 16, 32, 64

29. 1, 5, 9, 13, . . . , 29

30. 22, 4, 26, 8, . . . , 222

31. 0, 3, 8, 15, 24, 35

32. 21, 0, 1, 8, 27

33. 4, 29, 14, 219, 24

3 , 5 , . . . , 13 1, } 34. } } } 2 2 2 2

1, 2, 3, 4, . . . , 9 35. } } } } } 9 8 7 6 1

EXAMPLE 4

WRITING SUMMATION NOTATION Write the series using summation notation.

on p. 680 for Exs. 36–43

36. 7 1 10 1 13 1 16 1 19

37. 5 1 11 1 17 1 23 1 29

38. 21 1 1 1 3 1 5 1 7 1 . . .

39. 22 1 4 2 8 1 16 2 32 1 . . .

40. 3 1 10 1 17 1 24 1 31 1 . . .

11 1 1 1 1 1} 41. } } } 9 27 81 3

2131415161 7 1 1} 42. } } } } } } 5 7 6 8 9 10 4

43. 21 1 2 1 7 1 14 1 23 1 . . .

EXAMPLES 5 and 6

USING SUMMATION NOTATION Find the sum of the series.

on p. 681 for Exs. 44–57

44.

6

48.

∑ 2i

5

45.

i51

i51

6

5

∑ (5k 2 2)

49.

k53 35

52.

∑1

i51

682

∑ 7i

∑ (n2 2 1) n51

4

46.

∑n n51

Chapter 10 Sequences, Series, and the Binomial Theorem

4

47.

k51 8

50.

∑

2 i

}

i51

16

53.

∑ 3k2

6

51.

∑i

i51

∑

k51

25

54.

∑ n3 n50

18

55.

k k11

}

∑ n2 n51

56. ERROR ANALYSIS Describe and

correct the error in finding the sum of the series.

5

∑ (2i 1 3) 5 5 1 7 1 9 1 11 1 13 5 45 i50 20

∑ i?

57. ★ MULTIPLE CHOICE What is the sum of the series

i51

A 20

B 210

C 420

D 2870

58. ★ MULTIPLE CHOICE Which series

describes the total area of the first 10 triangles in the pattern shown at the right, in which the height stays constant but the base of successive triangles increases by 2? 10

A

∑ F 6 p (4 1 2i) G

For help with counterexamples see p. 770.

∑ F 6 p (2 1 2i) G

12

12

12 ...

6

4

10

B

i51

REVIEW LOGIC

12

8

10

i51

10

∑ 24(2i 2 1)

C

10

D

i51

∑ F 2(2i 2 1)p12 G

i51

CHALLENGE Tell whether the statement about summation notation is true or false. If the statement is true, prove it. If the statement is false, give a counterexample. n

59.

61.

n

∑ kai 5 k ∑ ai

i51

i51

n

n

60. n

∑ aibi 5 1 ∑ ai 21 ∑ bi 2

i51

i51

i51

62.

n

n

n

i51

i51

i51

∑ (ai 1 bi) 5 ∑ ai 1 ∑ bi n

n

i51

i51

∑ (ai)k 5 1 ∑ ai 2

k

PROBLEM SOLVING EXAMPLES 3 and 6 on pp. 679–681 for Exs. 63–64

63.

GEOMETRY For a regular n-sided polygon (n ≥ 3), the measure an of an interior angle is given by this formula:

180(n 2 2) an 5 } n

an

Write the first five terms of the sequence. Write a

rule for the sequence giving the total measure Tn of the interior angles in each regular n-sided polygon. Use the rule to find the total measure of the angles in the Guggenheim Museum skylight, which is a regular dodecagon.

Guggenheim Museum Skylight

For problem solving help, go to thinkcentral.com

64. ★ SHORT RESPONSE You want to save $500 for a school trip. You begin by

saving a penny on the first day. You plan to save an additional penny each day after that. For example, you will save 2 pennies on the second day, 3 pennies on the third day, and so on. How much money will you have saved after 100 days? How many days must you save to have saved $500? Explain how you used a series to find your answer. For problem solving help, go to thinkcentral.com

10.1 Define and Use Sequences and Series

683

65. TOWER OF HANOI In the puzzle called the Tower of Hanoi, the object is

to use a series of moves to take the rings from one peg and stack them in order on another peg. A move consists of moving exactly one ring, and no ring may be placed on top of a smaller ring. The minimum number an of moves required to move n rings is 1 for 1 ring, 3 for 2 rings, 7 for 3 rings, 15 for 4 rings, and 31 for 5 rings. Find a formula for the sequence. What is the minimum number of moves required to move 6 rings? 7 rings? 8 rings?

Start

Step 1

Step 2

Step 3

End

66. MULTI-STEP PROBLEM The mean distance dn (in astronomical units)

of each planet (except Neptune and Pluto) from the sun is approximated by the Titius-Bode rule, dn 5 0.3(2) n 2 2 1 0.4, where n is a positive integer representing the position of the planet from the sun. a. Evaluate The value of n is 4 for Mars. Use the Titius-Bode rule to

approximate the distance of Mars from the sun. b. Convert One astronomical unit is equal to about 149,600,000 kilometers.

How far is Mars from the sun in kilometers? c. Graph Graph the sequence given by the Titius-Bode rule. 67. ★ EXTENDED RESPONSE For a display at a sports store, you are stacking

soccer balls in a pyramid whose base is an equilateral triangle. The n(n 1 1) number an of balls per layer is given by an 5 } where n 5 1 2

represents the top layer.

a. How many balls are in the fifth layer? b. How many balls are in a stack with five layers? c. Compare the number of balls in a layer of a triangular pyramid with the

number of balls in the same layer of a square pyramid. 68. CHALLENGE Using the true statements from Exercises 59–62 on page 683 and

the special formulas on page 681, find a formula for the number of balls in the top n layers of the pyramid from Exercise 67.

FLORIDA SPIRAL REVIEW 69. A water fountain has an elliptical shape, with

MA.912.A.9.1

A 6

B 8

y

(2c, 0 )

water spouts located at each of its foci as shown. The fountain can be modeled by the equation 64x2 1 100y 2 2 6400 5 0 where x and y are in feet. What is the value of c? (9.4)

(c, 0 ) x

}

C 10

D 2Ï41

70. GRIDDED RESPONSE A regular square pyramid sculpture is 12 feet high.

It has a surface area of 360 square feet. An equation relating the surface

Î

}

MA.912.A.6.5

684

1 2 area to the side length of the base is 360 5 s2 1 2s } s 1 144 . What is 4 the side length of the base (in feet)? (6.6)

PRACTICE Lesson 10.1, Theorem p. 787 Chapter 10EXTRA Sequences, Series, andfor the Binomial

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FLORIDA

Graphing p g Calcu Cal Calculat Calculator Ca a alculator lcculato cula ullat ula ator t

Use after Lesson 10.1 ACTIVI ACTIVITY ACTIV AC ACTI A ACT CT C CTI TIIV TI TIV V Keystroke Help

10.1 Work with Sequences

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QUESTION

How can you use a graphing calculator to perform operations with sequences?

EXAMPLE

Preparation for MA.912.D.11.3 Find specified terms of arithmetic and geometric sequences.

Find, graph, and sum terms of a sequence

Use a graphing calculator to find the first eight terms of an 5 5n 2 3. Graph the sequence. Then find the sum of the first eight terms of the sequence.

STEP 1 Enter sequence

STEP 2 Calculate terms

Put the graphing calculator in sequence mode and dot mode. Enter the sequence. Note that the calculator uses u(n) rather than an .

Use the table feature to view the terms of the sequence. The first eight terms are 2, 7, 12, 17, 22, 27, 32, and 37.

n 1 2 3 4 5 n=1

nMin=1 u(n)=5n-3 u(nMin)= v(n)= v(nMin)= w(n)= w(nMin)=

u(n) 2 7 12 17 22

STEP 3 Graph sequence

STEP 4 Find sum of terms

Set the viewing window so that 1 ≤ n ≤ 8, 0 ≤ x ≤ 9, and 0 ≤ y ≤ 40. Graph the sequence. Use the trace feature to view the terms of the sequence.

Use the summation feature to find the sum of the first eight terms of the sequence. The screen shows that the sum is 156.

sum(seq(5n-3,n,1, 8)) 156

n=5 X=5

Y=22

PRACTICE Use a graphing calculator to (a) find the first ten terms of the sequence, (b) graph the sequence, and (c) find the sum of the first ten terms of the sequence. 1. an 5 4n 1 1 4. an 5 15 1 2n

2. an 5 3(n 1 2) 5. an 5 3 1 n

2

3. an 5 35 2 3n 6. an 5 2n 2 1 10.1 Define and Use Sequences and Series

685

FLORIDA

Analyze Arithmetic Sequences and Series

10.2

You defined and used general sequences and series.

Before

MA.912.D.11.1 Define arithmetic…sequences and series. MA.912.D.11.3 Find specified terms of arithmetic…sequences.

Now

Also MA.912.D.11.2, MA.912.D.11.4

So you can arrange a marching band, as in Ex. 64.

Why?

In an arithmetic sequence, the difference of consecutive terms is constant. This constant difference is called the common difference and is denoted by d.

Key Vocabulary • arithmetic

sequence • common

EXAMPLE 1

difference • arithmetic series

Identify arithmetic sequences

Tell whether the sequence is arithmetic. a. 24, 1, 6, 11, 16, . . .

b. 3, 5, 9, 15, 23, . . .

Solution Find the differences of consecutive terms. a. a2 2 a1 5 1 2 (24) 5 5

✓

b. a2 2 a1 5 5 2 3 5 2

a3 2 a2 5 6 2 1 5 5

a3 2 a2 5 9 2 5 5 4

a4 2 a 3 5 11 2 6 5 5

a4 2 a 3 5 15 2 9 5 6

a5 2 a 4 5 16 2 11 5 5

a5 2 a 4 5 23 2 15 5 8

c Each difference is 5, so the sequence is arithmetic.

c The differences are not constant, so the sequence is not arithmetic.

GUIDED PRACTICE

for Example 1

1. Tell whether the sequence 17, 14, 11, 8, 5, . . . is arithmetic. Explain why or

why not.

For Your Notebook

KEY CONCEPT Rule for an Arithmetic Sequence Algebra

The nth term of an arithmetic sequence with first term a1 and common difference d is given by: an 5 a1 1 (n 2 1)d

Example

The nth term of an arithmetic sequence with a first term of 2 and common difference 3 is given by: an 5 2 1 (n 2 1)3, or an 5 21 1 3n

686

Chapter 10 Sequences, Series, and the Binomial Theorem

EXAMPLE 2

Write a rule for the nth term

Write a rule for the nth term of the sequence. Then find a15. a. 4, 9, 14, 19, . . .

b. 60, 52, 44, 36, . . .

Solution a. The sequence is arithmetic with first term a1 5 4 and common

difference d 5 9 2 4 5 5. So, a rule for the nth term is:

AVOID ERRORS In the general rule for an arithmetic sequence, note that the common difference d is multiplied by n 2 1, not n.

an 5 a1 1 (n 2 1)d

Write general rule.

5 4 1 (n 2 1)5

Substitute 4 for a1 and 5 for d.

5 21 1 5n

Simplify.

The 15th term is a15 5 21 1 5(15) 5 74. b. The sequence is arithmetic with first term a1 5 60 and common

difference d 5 52 2 60 5 28. So, a rule for the nth term is: an 5 a1 1 (n 2 1)d

Write general rule.

5 60 1 (n 2 1)(28)

Substitute 60 for a1 and 28 for d.

5 68 2 8n

Simplify.

The 15th term is a15 5 68 2 8(15) 5 252.

EXAMPLE 3

Write a rule given a term and common difference

One term of an arithmetic sequence is a19 5 48. The common difference is d 5 3. a. Write a rule for the nth term.

b. Graph the sequence.

Solution a. Use the general rule to find the first term.

an 5 a1 1 (n 2 1)d

Write general rule.

a19 5 a1 1 (19 2 1)d

Substitute 19 for n.

48 5 a1 1 18(3)

Substitute 48 for a19 and 3 for d.

26 5 a1

Solve for a1.

So, a rule for the nth term is: an 5 a1 1 (n 2 1)d

Write general rule.

5 26 1 (n 2 1)3

Substitute 26 for a1 and 3 for d.

5 29 1 3n

Simplify.

an

b. Create a table of values for the sequence.

The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. n

1

2

3

4

5

6

an

26

23

0

3

6

9

3 1

n

10.2 Analyze Arithmetic Sequences and Series

687

EXAMPLE 4

Write a rule given two terms

Two terms of an arithmetic sequence are a 8 5 21 and a27 5 97. Find a rule for the nth term. Solution

ANOTHER WAY For an alternative method for solving the problem in Example 4, turn to page 694 for the Problem Solving Workshop.

STEP 1

Write a system of equations using an 5 a1 1 (n 2 1)d and substituting

27 for n (Equation 1) and then 8 for n (Equation 2). a27 5 a1 1 (27 2 1)d

97 5 a1 1 26d

Equation 1

a 8 5 a1 1 (8 2 1)d

21 5 a1 1 7d

Equation 2

76 5

Subtract.

STEP 2 Solve the system.

19d

45d

STEP 3 Find a rule for an.

✓

GUIDED PRACTICE

Solve for d.

97 5 a1 1 26(4)

Substitute for d in Equation 1.

27 5 a1

Solve for a1.

an 5 a1 1 (n 2 1)d

Write general rule.

5 27 1 (n 2 1)4

Substitute for a1 and d.

5 211 1 4n

Simplify.

for Examples 2, 3, and 4

Write a rule for the nth term of the arithmetic sequence. Then find a20. 2. 17, 14, 11, 8, . . .

3. a11 5 257, d 5 27

4. a7 5 26, a16 5 71

ARITHMETIC SERIES The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series. The sum of the first n terms of an arithmetic series is denoted by Sn . To find a rule for Sn, you can write Sn in two different ways and add the results.

Sn 5 a1

1 (a1 1 d) 1 (a1 1 2d) 1 . . . 1 an

Sn 5 an

1 (an 2 d) 1 (an 2 2d) 1 . . . 1 a1

2Sn 5 (a1 1 an ) 1 (a1 1 an ) 1 (a1 1 an ) 1 . . . 1 (a1 1 an ) You can conclude that 2Sn 5 n(a1 1 an ), which leads to the following result.

For Your Notebook

KEY CONCEPT The Sum of a Finite Arithmetic Series The sum of the first n terms of an arithmetic series is:

1

a 1a

1 n Sn 5 n }

2

2

In words, Sn is the mean of the first and nth terms, multiplied by the number of terms.

688

Chapter 10 Sequences, Series, and the Binomial Theorem

★

EXAMPLE 5

Standardized Test Practice 20

What is the sum of the arithmetic series

∑ (3 1 5i)?

i51

A 103 CLASSIFY SERIES You can verify that the series in Example 5 is arithmetic by evaluating 3 1 5i for the first few values of the index i. The resulting terms are 8, 13, 18, 23, . . . , which have a common difference of 5.

B 111

C 1110

D 2220

Solution a1 5 3 1 5(1) 5 8

Identify first term.

a20 5 3 1 5(20) 5 103

Identify last term.

8 1 103 S20 5 20 }

Write rule for S20, substituting 8 for a1 and 103 for a20.

1

2

2

5 1110

Simplify.

c The correct answer is C. A B C D

EXAMPLE 6

Use an arithmetic sequence and series in real life

HOUSE OF CARDS You are making a

house of cards similar to the one shown. first row

a. Write a rule for the number of cards

in the nth row if the top row is row 1. b. What is the total number of cards if

the house of cards has 14 rows?

Solution a. Starting with the top row, the numbers of cards in the rows are 3, 6, 9,

12, . . . . These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is: an 5 a1 1 (n 2 1)d

Write general rule.

5 3 1 (n 2 1)3

Substitute 3 for a1 and 3 for d.

5 3n

Simplify.

b. Find the sum of an arithmetic series with first term a1 5 3 and last

term a14 5 3(14) 5 42.

a1 1 a14 3 1 42 5 315 Total number of cards 5 S14 5 14 } 5 14 }

1

2

2

1

2

2

"MHFCSB Go to thinkcentral.com

✓

GUIDED PRACTICE

for Examples 5 and 6 12

5. Find the sum of the arithmetic series

∑ (2 1 7i).

i51

6. WHAT IF? In Example 6, what is the total number of cards if the house of

cards has 8 rows? 10.2 Analyze Arithmetic Sequences and Series

689

10.2

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS18 for Exs. 15, 41, and 65

★

MA.912.D.11.1, MA.912.D.11.3

5 STANDARDIZED TEST PRACTICE Exs. 2, 29, 39, 52, and 68 5 MULTIPLE REPRESENTATIONS Ex. 66

SKILL PRACTICE 1. VOCABULARY Copy and complete: The constant difference between

consecutive terms of an arithmetic sequence is called the ? . 2. ★ WRITING Explain the difference between an arithmetic sequence and an

arithmetic series. EXAMPLE 1

IDENTIFYING ARITHMETIC SEQUENCES Tell whether the sequence is arithmetic.

on p. 686 for Exs. 3–11

Explain why or why not. 3. 1, 22, 25, 28, 211, . . .

4. 16, 14, 11, 6, 3, . . .

5. 5, 14, 23, 32, 41, . . .

6. 210, 27, 25, 22, 0, . . .

7. 0.5, 1, 1.5, 2, 2.5, . . .

8. 20, 10, 5, 2.5, 1.25, . . .

7, 5, 3, 23, 25, . . . 9. } } } } } 4 4 4 4 4

2 , 4 , 8 , 16 , . . . 1, } 10. } } } } 7 7 7 7 7

5 , 21, 1 , 2, 7 , . . . 11. 2} } } 2 2 2

EXAMPLE 2

WRITING RULES Write a rule for the nth term of the arithmetic sequence. Then

on p. 687 for Exs. 12–22

find a20. 12. 1, 4, 7, 10, 13, . . .

13. 5, 11, 17, 23, 29, . . .

14. 8, 21, 34, 47, 60, . . .

15. 23, 21, 1, 3, 5, . . .

16. 6, 2, 22, 26, 210, . . .

17. 25, 14, 3, 28, 219, . . .

2 , 4 , 2, 8 , . . . 18. 0, } } } 3 3 3

5 , 4 , 1, 2 , . . . 19. 2, } } } 3 3 3

20. 1.5, 3.6, 5.7, 7.8, 9.9, . . .

ERROR ANALYSIS Describe and correct the error in writing the rule for the nth

term of the arithmetic sequence 37, 24, 11, 22, 215, . . . . 21.

22.

Use a1 5 37 and d 5 213.

The first term is 37 and the common difference is 213.

an 5 a1 1 nd

an 5 213 1 (n 2 1)(37)

an 5 37 1 n(213)

an 5 250 1 37n

an 5 37 2 13n EXAMPLE 3 on p. 687 for Exs. 23–29

WRITING RULES Write a rule for the nth term of the arithmetic sequence. Then graph the first six terms of the sequence.

23. a16 5 52, d 5 5

24. a 6 5 216, d 5 9

25. a 4 5 96, d 5 214

26. a12 5 23, d 5 27

7 27. a10 5 30, d 5 } 2

1, d 5 21 28. a11 5 } } 2 2

29. ★ MULTIPLE CHOICE For a certain arithmetic sequence, a 30 5 57 and d 5 4.

What is a rule for the nth term of the sequence?

690

A an 5 263 2 4n

B an 5 259 2 4n

C an 5 263 1 4n

D an 5 259 1 4n

Chapter 10 Sequences, Series, and the Binomial Theorem

EXAMPLE 4 on p. 688 for Exs. 30–39

WRITING RULES Write a rule for the nth term of the arithmetic sequence that has the two given terms.

30. a 4 5 31, a10 5 85

31. a 6 5 39, a14 5 79

32. a 3 5 22, a17 5 40

33. a 8 5 210, a20 5 258

34. a 9 5 89, a15 5 137

35. a2 5 17, a11 5 35

36. a7 5 4, a12 5 29

37. a5 5 15, a 9 5 24

38. a 6 5 0, a11 5 22

39. ★ MULTIPLE CHOICE For a certain arithmetic sequence, a 6 5 26 and

a13 5 248. What is a rule for the nth term of the sequence? A an 5 18 1 6n

B an 5 30 2 6n

C an 5 26 1 24n

D an 5 236 2 6n

EXAMPLE 5

FINDING SUMS Find the sum of the arithmetic series.

on p. 689 for Exs. 40–48

40.

10

8

∑ (1 1 3i)

i51

42.

i51

∑ (29 1 11i)

i51

14

∑ (72 2 6i)

44.

45.

i53

46. 2 1 6 1 10 1 . . . 1 58

∑ (14 2 6i)

i51

9

22

43.

18

∑ (23 2 2i)

41.

∑ (254 1 9i)

i55

47. 21 1 4 1 9 1 . . . 1 34

48. 44 1 37 1 30 1 . . . 1 2

USING GRAPHS Write a rule for the sequence whose graph is shown.

an

49.

an

50. (4, 17)

22

(3, 12)

1

(1, 2)

5

n

22

(2, 24) (3, 27)

(2, 7) 3

an

51. (1, 21)

(4, 210)

n

1

(1, 23) (2, 25) (3, 27) (4, 29)

n

52. ★ WRITING Compare the graph of an 5 3n 1 2, where n is a positive integer,

with the graph of f (x) 5 3x 1 2, where x is a real number. Discuss how the graph of an arithmetic sequence is similar to and different from the graph of a linear function.

REASONING Tell whether the statement is true or false. Explain your answer.

53. If the common difference of an arithmetic series is doubled while the first

term and number of terms in the series remain unchanged, then the sum of the series is doubled. 54. If the numbers a, b, and c are the first three terms of an arithmetic sequence,

then b is half the sum of a and c. SOLVING EQUATIONS Find the value of n. n

55.

58.

∑ (25 1 7i) 5 486

n

56.

∑ (10 2 3i) 5 228

i51

i51

n

n

∑ (5 2 5i) 5 250

i51

59.

∑ (23 2 4i) 5 2507

i53

n

57.

∑ (58 2 8i) 5 21150

i51 n

60.

∑ (7 1 12i) 5 455

i55

61. REASONING Find the sum of all positive odd integers less than 300. 62. CHALLENGE The numbers 3 2 x, x, and 1 2 3x are the first three terms in an

arithmetic sequence. Find the value of x and the next term in the sequence.

10.2 Analyze Arithmetic Sequences and Series

691

PROBLEM SOLVING EXAMPLE 6

63. HONEYCOMBS Domestic bees make their honeycomb

by starting with a single hexagonal cell, then forming ring after ring of hexagonal cells around the initial cell, as shown. The numbers of cells in successive rings form an arithmetic sequence.

on p. 689 for Exs. 63–65

a. Write a rule for the number of cells in the nth ring. b. What is the total number of cells in the honeycomb

Initial cell

1 ring

2 rings

after the 9th ring is formed? (Hint: Do not forget to count the initial cell.) For problem solving help, go to thinkcentral.com

64. MARCHING BAND A marching band is arranged in 7 rows. The first row has

3 band members, and each row after the first has 2 more band members than the row before it. Write a rule for the number of band members in the nth row. Then find the total number of band members. For problem solving help, go to thinkcentral.com

65. SCULPTURE Sol LeWitt’s sculpture Four-Sided Pyramid in the National

Gallery of Art Sculpture Garden is made of concrete blocks. As shown in the diagram, each layer has 8 more visible blocks than the layer in front of it.

a. Write a rule for the number of visible blocks in the nth layer where n 5 1

represents the front layer. b. When you view the pyramid from one corner, a total of 12 layers are

visible. How many of the pyramid’s blocks are visible? 66.

MULTIPLE REPRESENTATIONS The distance D (in feet) that an object falls in t seconds can be modeled by D(t) 5 16t 2.

a. Making a Table Let d(n) represent the distance the object falls in the

nth second. Make a table of values showing d(1), d(2), d(3), and d(4). (Hint: The distance d(1) that the object falls in the first second is D(1) 2 D(0).) b. Writing a Rule Write a rule for the sequence of distances given by d(n). c. Drawing a Graph Graph the sequence from part (b). 67. ENTERTAINMENT During a high school spirit week, students dress up

in costumes. A cash prize is given each day to the student with the best costume. The organizing committee has $1000 to give away over five days. The committee wants to increase the amount of the prize by $50 each day. How much should the committee give away on the first day?

692

5 WORKED-OUT SOLUTIONS

Chapter 12 Sequences Series starting on and p. WS1

★

5 STANDARDIZED TEST PRACTICE

5 MULTIPLE REPRESENTATIONS

68. ★ EXTENDED RESPONSE A paper towel manufacturer sells paper towels

rolled onto cardboard dowels. The thickness of the paper is 0.004 inch. The diameter of a dowel is 2 inches, and the total diameter of a roll is 5 inches. n

dn (in.)

ln (in.)

1

2

2π

2

?

?

3

?

?

4

?

?

1.5 in.

2 in.

1.5 in.

5 in.

a. Calculate Let n be the number of times the paper towel is wrapped

around the dowel, let dn be the diameter of the roll just before the nth wrap, and let ln be the length of paper added in the nth wrap. Copy and complete the table. b. Model What kind of sequence is l1, l2, l3, l4, . . . ? Write a rule for the

nth term of the sequence.

c. Apply Find the number of times the paper must be wrapped around

the dowel to create a roll with a 5 inch diameter. Use your answer and the rule from part (b) to find the length of paper in a roll with a 5 inch diameter. d. Interpret Suppose a roll with a 5 inch diameter costs $1.50. How much

would you expect to pay for a roll with a 7 inch diameter whose dowel also has a diameter of 2 inches? Explain your reasoning and any assumptions you make. 69. CHALLENGE A theater has n rows of seats, and each row has d more seats

than the row in front of it. There are x seats in the last (nth) row and a total of y seats in the entire theater. How many seats are in the front row of the theater? Write your answer in terms of n, x, and y.

FLORIDA SPIRAL REVIEW 70. GRIDDED RESPONSE You can make a solar hot

MA.912.A.9.1

dog cooker by shaping foil-lined cardboard into a parabolic trough and passing a wire through the focus of each end piece. For the trough shown, how far from the bottom, in inches, should the wire be placed? (9.2)

71. Cell phone towers are built every 40 miles

40 mi

40 mi

along a rural highway. The band along the highway that is always in range of one or more towers is 30 miles wide, as shown. What is the equation of a circle that represents the range of a cell phone tower if the tower is chosen as the origin? (9.3) MA.912.G.6.6

A x2 1 y 2 5 225

B x2 1 y 2 5 400

30 mi

C x2 1 y 2 5 625

D x2 1 y 2 5 900

EXTRA PRACTICE for Lesson 10.2, p. 787 12.2ONLINE Go toSequences thinkcentral.com AnalyzeQUIZ Arithmetic and Series

693

FLORIDA A

Problem Problem P Solving Solvi Solv So S Solvin olving olv llving vin v vi iin ng ng

MA.912.A.3.10 Write an equation of a line given…two points on the line…

Use with Lesson 10.2 WORKSHOP WORKS WOR W WO O OR ORKSHOP R

Another Way to Solve Example 4, p. 688 MULTIPLE REPRESENTATIONS In Example 4 on page 688, you solved a system of equations to write a rule for the nth term of an arithmetic sequence. You can also use two given terms of a sequence as two points of a line and write a rule using point-slope form. This can be done because arithmetic sequences are linear functions with an the common difference, d, being the slope, the relative position of each term being the x-value, and the value of each term being the corresponding y-value.

For instance, consider the graph of an 5 211 1 4n, shown at the right. Each time a domain value n increases by 1, its range value an increases by 4. So, the rise divided by the run is equal to the constant slope, 4. Note that ordered pairs are written in the form (n, an ).

PROBLEM

METHOD

2 1

n

Two terms of an arithmetic sequence are a 8 5 21 and a27 5 97. Find a rule for the nth term.

Using Point-Slope Form You are given the terms a 8 5 21 and a27 5 97. Use the points (x1, y1) 5 (8, 21) and (x2, y 2) 5 (27, 97) to find the slope. y 2y

97 2 21 76 2 1 m5 } x 2x 5} 5}54 2

1

27 2 8

19

Use point-slope form with y 5 an and x 5 n and either given term to write a rule for the sequence. Use (x1, y1) 5 (8, 21). an – y1 5 m(n 2 x1)

Use point-slope form with y = an and x = n.

an – 21 5 4(n 2 8)

Substitute 4 for m, 8 for x1, and 21 for y1.

an – 21 5 4n 2 32

Distributive property

an 5 4n 2 11

Write in slope-intercept form.

A rule for the sequence is an 5 4n 2 11 or an 5 211 1 4n.

P R AC T I C E Use point-slope form to write a rule for the nth term of the arithmetic sequence.

694

1. a3 5 16, a 4 5 20

2. a10 5 32, a12 5 48

3. a 4 5 93, a 8 5 65

4. a2 5 10, a6 5 8

5. There are 25 rows of seats at an outdoor theater. The first row has 20 seats and the last row has 44 seats. Each row is the same number of seats longer than the row before it. Use the pointslope form to write a rule for the number of seats in the nth row.

Chapter 10 Sequences, Series, and the Binomial Theorem

FLORIDA

10.3

Analyze Geometric Sequences and Series You defined and used arithmetic sequences and series.

Before

MA.912.D.11.1 Define…geometric sequences and series. MA.912.D.11.3 Find specified terms of…geometric sequences. Also MA.912.D.11.2, MA.912.D.11.4

Now

So you can solve problems about soccer tournaments, as in Ex. 58.

Why?

Key Vocabulary • geometric sequence • common ratio • geometric series

In a geometric sequence, the ratio of any term to the previous term is constant. This constant ratio is called the common ratio and is denoted by r.

EXAMPLE 1

Identify geometric sequences

Tell whether the sequence is geometric. a. 4, 10, 18, 28, 40, . . .

b. 625, 125, 25, 5, 1, . . .

Solution To decide whether a sequence is geometric, find the ratios of consecutive terms. a3

a 10 5 5 a. }2 5 } } 4 2 a1

18 10

a4

9 5

}5}5}

a2

28 18

14 9

}5}5}

a3

a5

40 28

10 7

}5}5}

a4

c The ratios are different, so the sequence is not geometric. a3

a 125 5 1 b. }2 5 } } 5 625 a1

25 125

a4

1 5

}5}5}

a2

5 25

1 5

}5}5}

a3

a5

1 5

}5}

a4

1 , so the sequence is geometric. c Each ratio is } 5

✓

GUIDED PRACTICE

for Example 1

Tell whether the sequence is geometric. Explain why or why not. 1. 81, 27, 9, 3, 1, . . .

2. 1, 2, 6, 24, 120, . . .

3. 24, 8, 216, 32, 264, . . .

For Your Notebook

KEY CONCEPT Rule for a Geometric Sequence Algebra

The nth term of a geometric sequence with first term a1 and common ratio r is given by: an 5 a1r n 2 1

Example

The nth term of a geometric sequence with a first term of 3 and common ratio 2 is given by: an 5 3(2) n 2 1

10.3 Analyze Geometric Sequences and Series

695

EXAMPLE 2

Write a rule for the nth term

Write a rule for the nth term of the sequence. Then find a7. a. 4, 20, 100, 500, . . .

b. 152, 276, 38, 219, . . .

Solution a. The sequence is geometric with first term a1 5 4 and common ratio 20 5 5. So, a rule for the nth term is: r5} 4 AVOID ERRORS In the general rule for a geometric sequence, note that the exponent is n 2 1, not n.

an 5 a1r n 2 1

Write general rule.

5 4(5) n 2 1

Substitute 4 for a1 and 5 for r.

The 7th term is a7 5 4(5)7 2 1 5 62,500. b. The sequence is geometric with first term a1 5 152 and common ratio 276 5 2 1 . So, a rule for the nth term is: r5} } 152 2

an 5 a1r n 2 1

Write general rule.

1 5 152 1 2} 2

n21

1 2

Substitute 152 for a1 and 2} for r.

2

721

1 The 7th term is a7 5 152 1 2} 2 2

EXAMPLE 3

19 . 5} 8

Write a rule given a term and common ratio

One term of a geometric sequence is a 4 5 12. The common ratio is r 5 2. a. Write a rule for the nth term.

b. Graph the sequence.

Solution a. Use the general rule to find the first term.

an 5 a1r n 2 1 a 4 5 a1r

Write general rule.

421

12 5 a1(2)

Substitute 4 for n.

3

Substitute 12 for a4 and 2 for r.

1.5 5 a1

Solve for a1.

So, a rule for the nth term is: an 5 a1r n 2 1 5 1.5(2)

Write general rule.

n21

Substitute 1.5 for a1 and 2 for r.

b. Create a table of values for the sequence. The graph

an

of the first 6 terms of the sequence is shown. Notice that the points lie on an exponential curve. This is true for any geometric sequence with r > 0. n

1

2

3

4

5

6

an

1.5

3

6

12

24

48

"MHFCSB Go to thinkcentral.com

696

Chapter 10 Sequences, Series, and the Binomial Theorem

10 1

n

EXAMPLE 4

Write a rule given two terms

Two terms of a geometric sequence are a3 5 248 and a 6 5 3072. Find a rule for the nth term. Solution

STEP 1

Write a system of equations using an 5 a1r n 2 1 and substituting 3 for n

(Equation 1) and then 6 for n (Equation 2). a 3 5 a1r 3 2 1

248 5 a1r 2

Equation 1

a 6 5 a1r 6 2 1

3072 5 a1r 5

Equation 2

STEP 2 Solve the system.

248 r

5 a1 } 2

Solve Equation 1 for a1.

248 (r 5) 3072 5 } 2

Substitute for a1 in Equation 2.

3072 5 248r 3

Simplify.

r

24 5 r

Solve for r.

248 5 a1(24)

2

23 5 a1

STEP 3 Find a rule for an.

an 5 a1r

n21

an 5 23(24) n 2 1

✓

GUIDED PRACTICE

Substitute for r in Equation 1. Solve for a1. Write general rule. Substitute for a1 and r.

for Examples 2, 3, and 4

Write a rule for the nth term of the geometric sequence. Then find a 8. 4. 3, 15, 75, 375, . . .

5. a 6 5 296, r 5 2

6. a2 5 212, a 4 5 23

GEOMETRIC SERIES The expression formed by adding the terms of a geometric sequence is called a geometric series. The sum of the first n terms of a geometric series is denoted by Sn . You can develop a rule for Sn as follows.

Sn 5 a1 1 a1r 1 a1r 2 1 a1r 3 1 . . . 1 a1r n 2 1 2rSn 5

2 a1r 2 a1r 2 2 a1r 3 2 . . . 2 a1r n 2 1 2 a1r n

Sn (1 2 r) 5 a1 1 0 1 0 1 0 1 . . . 1

0

2 a1r n

So, Sn (1 2 r) 5 a1(1 2 r n). If r Þ 1, you can divide each side of this equation by 1 2 r to obtain the following rule for Sn .

For Your Notebook

KEY CONCEPT The Sum of a Finite Geometric Series

The sum of the first n terms of a geometric series with common ratio r Þ 1 is: 1 2 rn Sn 5 a1 }

1 12r 2

10.3 Analyze Geometric Sequences and Series

697

EXAMPLE 5

Find the sum of a geometric series 16

Find the sum of the geometric series

∑ 4(3)i 2 1.

i51

a1 5 4(3)

121

54

Identify first term.

r53

Identify common ratio.

1 2 r16 S16 5 a1 }

1

12r

2

Write rule for S16.

1 2 316 54 }

2

Substitute 4 for a1 and 3 for r.

1

123

5 86,093,440

Simplify.

c The sum of the series is 86,093,440.

EXAMPLE 6

Use a geometric sequence and series in real life

MOVIE REVENUE In 1990, the total box office

revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year. a. Write a rule for the total box office

revenue an (in billions of dollars) in terms of the year. Let n 5 1 represent 1990. b. What was the total box office revenue

at U.S. movie theaters for the entire period 1990–2003? Solution a. Because the total box office revenue increased by the same percent

each year, the total revenues from year to year form a geometric sequence. Use a1 5 5.02 and r 5 1 1 0.059 5 1.059 to write a rule for the sequence. an 5 5.02(1.059) n 2 1

Write a rule for an .

b. There are 14 years in the period 1990–2003, so find S14. 14 2 r 14 5 5.02 1 2 (1.059) S14 5 a1 1} } ø 105

1

12r

2

1

2

1 2 1.059

c The total movie box office revenue for the period 1990–2003 was about $105 billion.

✓

GUIDED PRACTICE

for Examples 5 and 6 8

7. Find the sum of the geometric series

∑ 6(22)i 2 1.

i51

8. MOVIE REVENUE Use the rule in part (a) of Example 6 to estimate the total

box office revenue at U.S. movie theaters in 2000.

698

Chapter 10 Sequences, Series, and the Binomial Theorem

10.3

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS18 for Exs. 19, 49, and 59

★

MA.912.D.11.1, MA.912.D.11.3

5 STANDARDIZED TEST PRACTICE Exs. 2, 27, 54, 55, and 59 5 MULTIPLE REPRESENTATIONS Ex. 60

SKILL PRACTICE 1. VOCABULARY Copy and complete: The constant ratio of consecutive terms

in a geometric sequence is called the ? . 2. ★ WRITING How can you determine whether a sequence is geometric? EXAMPLE 1

IDENTIFYING GEOMETRIC SEQUENCES Tell whether the sequence is geometric.

on p. 695 for Exs. 3–14

Explain why or why not. 3. 1, 4, 8, 16, 32, . . .

4. 4, 16, 64, 256, 1024, . . .

1, . . . 5. 216, 36, 6, 1, } 6

2 , 4 , 8 , 16 , . . . 1, } 6. } } } } 3 3 3 3 3

3 , 2, 5 , . . . 1 , 1, } 7. } } 2 2 2

1, 3, 2 3 , 1 , 2 3 , . . . 8. 2} } } } } 4 8 16 32 64

9. 10, 5, 2.5, 1.25, 0.625, . . . 12. 0.2, 0.6, 1.8, 5.4, 16.2, . . .

10. 23, 26, 12, 24, 248, . . .

11. 24, 12, 236, 108, 2324, . . .

13. 25, 10, 20, 40, 80, . . .

14. 0.75, 1.5, 2.25, 3, 3.75, . . .

EXAMPLE 2

WRITING RULES Write a rule for the nth term of the geometric sequence. Then

on p. 696 for Exs. 15–27

find a7. 15. 1, 24, 16, 264, . . .

16. 6, 18, 54, 162, . . .

17. 4, 24, 144, 864, . . .

18. 7, 235, 175, 2875, . . .

3 , 9 , 27 , . . . 19. 2, } } } 2 8 32

6 , 12 , 2 24 , . . . 20. 3, 2} } } 5 25 125

21. 4, 2, 1, 0.5, . . .

22. 20.3, 0.6, 21.2, 2.4, . . .

23. 22, 20.8, 20.32, 20.128, . . .

24. 7, 24.2, 2.52, 21.512, . . .

25. 5, 214, 39.2, 2109.76, . . .

26. 120, 180, 270, 405, . . .

27. ★ MULTIPLE CHOICE What is a rule for the nth term of the geometric

sequence 5, 20, 80, 320, . . . ?

EXAMPLE 3 on p. 696 for Exs. 28–38

A an 5 5(2) n 2 1

B an 5 5(4) n 2 1

C an 5 5(24) n 2 1

D an 5 5(22) n 2 1

WRITING RULES Write a rule for the nth term of the geometric sequence. Then graph the first six terms of the sequence.

28. a1 5 5, r 5 3

29. a1 5 22, r 5 6

30. a2 5 6, r 5 2

1 31. a2 5 15, r 5 } 2

1 32. a5 5 1, r 5 } 8

1 33. a 4 5 212, r 5 2} 4

34. a 3 5 75, r 5 5

35. a2 5 8, r 5 4

36. a 4 5 500, r 5 5

ERROR ANALYSIS Describe and correct the error in writing the rule for the nth term of the geometric sequence for which a1 5 3 and r 5 2.

37.

an 5 a1rn an 5 3(2) n

38.

an 5 ra1n 2 1 an 5 2(3) n 2 1

10.3 Analyze Geometric Sequences and Series

699

EXAMPLE 4 on p. 697 for Exs. 39–47

WRITING RULES Write a rule for the nth term of the geometric sequence that has the two given terms.

39. a1 5 3, a 3 5 12

40. a1 5 1, a5 5 625

1 , a 5 216 41. a1 5 2} 4 4

42. a 3 5 10, a 6 5 270

43. a2 5 240, a 4 5 210

44. a2 5 224, a5 5 1536

45. a 4 5 162, a7 5 4374

7, a 5 7 46. a 3 5 } } 4 5 16

243 47. a 4 5 6, a7 5 } 8

EXAMPLE 5

FINDING SUMS Find the sum of the geometric series.

on p. 698 for Exs. 48–54

48.

10

51.

8

∑ 5(2)i 2 1

i51

i51

6

12

∑ 4 1 }14 2

i21

52.

i51

7

∑ 6(4)i 2 1

49.

∑ 8 1 }32 2

50.

∑ 121 2}12 2

i

i50 10

i21

53.

i51

∑ (24)i

i50 9

54. ★ MULTIPLE CHOICE What is the sum of the geometric series

∑ 2(3)i 2 1?

i51

A 19,680

B 19,681

C 19,682

D 19,683

55. ★ OPEN-ENDED MATH Write a geometric series with 5 terms such that the

sum of the series is 100. (Hint: Choose a value of r and then find a1.) 56. CHALLENGE Using the rule for the sum of a finite geometric series, write

each polynomial as a rational expression. a. 1 1 x 1 x 2 1 x 3 1 x4

b. 3x 1 6x 3 1 12x5 1 24x 7

PROBLEM SOLVING EXAMPLE 6 on p. 698 for Exs. 57–59

57. SKYDIVING In a skydiving formation

with R rings, each ring after the first has twice as many skydivers as the preceding ring. The formation for R 5 2 is shown. a. Let an be the number of

First ring

skydivers in the nth ring. Find a rule for an .

b. Find the total number

of skydivers if there are R 5 4 rings.

Second ring

For problem solving help, go to thinkcentral.com

58. SOCCER A regional soccer tournament has 64 participating teams. In the

first round of the tournament, 32 games are played. In each successive round, the number of games played decreases by one half. a. Find a rule for the number of games played in the nth round. For what

values of n does your rule make sense? b. Find the total number of games played in the regional soccer

tournament. For problem solving help, go to thinkcentral.com

700

5 WORKED-OUT SOLUTIONS

Chapter 12 Sequences Series starting on and p. WS1

★

5 STANDARDIZED TEST PRACTICE

5 MULTIPLE REPRESENTATIONS

59. ★ SHORT RESPONSE A binary search technique used on a computer involves

jumping to the middle of an ordered list of data (such as an alphabetical list of names) and deciding whether the item being searched for is there. If not, the computer decides whether the item comes before or after the middle. Half of the list is ignored on the next pass, and the computer jumps to the middle of the remaining list. This is repeated until the item is found. )NITIALDATA

)TEMSTOBESEARCHED

&IRSTPASS

3ECONDPASS

4HIRDPASS

)TEMSTOBEIGNORED

a. Find a rule for the number of items remaining after the nth pass

through an ordered list of 1024 items. b. In the worst case, the item to be found is the only one left in the list after

n passes through the list. What is the worst-case value of n for a binary search of a list with 1024 items? Explain. 60.

MULTIPLE REPRESENTATIONS Two companies, company A and company B, offer the same starting salary of $20,000 per year. Company A gives a raise of $1000 each year. Company B gives a raise of 4% each year.

a. Writing Rules Write rules giving the salaries an and bn in the nth year at

companies A and B, respectively. Tell whether the sequence represented by each rule is arithmetic, geometric, or neither.

b. Drawing Graphs Graph each sequence in the same coordinate plane. c. Finding Sums For each company, find the sum of wages earned during

the first 20 years of employment. d. Using Technology Use a graphing calculator or spreadsheet to find after

how many years the total amount earned at company B is greater than the total amount earned at company A. 61. CHALLENGE Every January 1, you deposit $2000 in an account that pays 5%

annual interest. You make a total of 30 deposits. How much money do you have in your account immediately after you make your last deposit?

FLORIDA SPIRAL REVIEW 62. GRIDDED RESPONSE Anita, Ben, and Carlos

MA.912.A.3.15

download the numbers of movies, songs, and TV show episodes shown. Anita spends $33, Ben spends $39, and Carlos spends $43. What is the cost (in dollars) to download a movie? (3.5)

Movies

Songs

TV shows

Anita

2

3

5

Ben

1

9

10

Carlos

3

13

0

63. GRIDDED RESPONSE Bacteria are introduced into a culture in a lab. The

MA.912.A.8.5

population of bacteria in the culture (in millions) is modeled by the function f(x) 5 5 p 2 0.75x 1 3 where x is the number of hours since the bacteria were introduced. If the number of bacteria in the culture is 20.5 billion, how many hours has it been since the bacteria were introduced? Round your answer to the nearest hour. (7.6)

EXTRA PRACTICE for Lesson 10.3, p. 787 12.3ONLINE Go toSequences thinkcentral.com AnalyzeQUIZ Geometric and Series

701

Expl Exploring E pl ring i g Algebra Algeb Al Algebra Algebr geeeb g gebr brra

FLORIDA

ACTIV ACTIVITY A AC ACT CT C CTIV CTI TI

Use before Lesson 10.4

10.4 Investigating an Infinite

MA.912.D.11.4 …find sums of infinite convergent geometric series…

Geometric Series

M AT E R I A L S • scissors • paper

QUESTION

What is the sum of an infinite geometric series?

You can illustrate an infinite geometric series by cutting a piece of paper into smaller and smaller pieces.

EXPLORE

Model an infinite geometric series

Start with a rectangular piece of paper. Define its area to be 1 square unit.

STEP 1 Cut paper in half

Fold the paper in half and cut along the fold. Place one half on a desktop and hold the remaining half.

STEP 2 Cut paper again

STEP 3 Repeat steps

Repeat Steps 1 and 2 until you find it too difficult to fold and cut the piece of paper you are holding.

Fold the piece of paper you are holding in half and cut along the fold. Place one half on the desktop and hold the remaining half.

STEP 4 Find areas The first piece of paper on the desktop has an area of 1 1 } square unit. The second piece has an area of } square unit. Write the areas 2 4

of the next three pieces of paper. Explain why these areas form a geometric sequence.

STEP 5 Make a table Copy and complete the table by recording the number of pieces of paper on the desktop and the combined area of the pieces at each step.

DR AW CONCLUSIONS

Number of pieces

1

2

3

4

...

Combined area

}

1 2

}1}5?

1 4

?

?

...

1 2

Use your observations to complete these exercises

1. Based on your table, what number does the combined area of the pieces of

paper appear to be approaching? 2. Using the formula for the sum of a finite geometric series, write and simplify

a rule for the combined area An of the pieces of paper after n cuts. What happens to An as n → `? Justify your answer mathematically.

702

Chapter 10 Sequences, Series, and the Binomial Theorem

FLORIDA

10.4 Before Now Why?

Key Vocabulary • partial sum

Find Sums of Infinite Geometric Series You found the sums of finite geometric series. MA.912.D.11.4 Find partial sums of arithmetic and geometric series, and find sums of infinite convergent geometric series. Use Sigma notation… Also MA.912.D.11.2 So you can analyze a fractal, as in Ex. 42.

The sum Sn of the first n terms of an infinite series is called a partial sum. The partial sums of an infinite geometric series may approach a limiting value. If so, the series is said to be convergent.

EXAMPLE 1

Find partial sums

1 1 1 1 1 1 1 1 1 1 . . . . Find and Consider the infinite geometric series } } } } } 2

4

8

16

32

graph the partial sums Sn for n 5 1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases. Solution

0.8

1 1 1 5 0.75 S2 5 } } 2

0.6

4

0.4

1 1 1 1 1 < 0.88 S3 5 } } } 2

4

Sn

1.0

1 5 0.5 S1 5 } 2

8

0.2

1 1 1 1 1 1 1 ø 0.94 S4 5 } } } } 2 4 8 16

1

2

1 1 1 1 1 1 1 1 1 ø 0.97 S5 5 } } } } } 2 4 8 16 32

3

4

5n

"MHFCSB

From the graph, Sn appears to approach 1 as n increases.

Go to thinkcentral.com

SUMS OF INFINITE SERIES In Example 1, you can understand why Sn approaches 1

as n increases by considering the rule for Sn : n

1

1 1 2 1} 2

n

2

2 12r 5 1 1 n Sn 5 a1 } } } 512 } 1 12r 2 12} 2 2

1

2

1 2

n

1 approaches 0, so S approaches 1. Therefore, 1 is defined to As n increases, 1 } 2 n 2

be the sum of the infinite geometric series in Example 1. More generally, as n increases for any infinite geometric series with common ratio r between 21 and

1 2 r n ø a 1 2 0 5 a1 . 1, the value of Sn 5 a1 } } 1 }

1 12r 2

112r2

12r

10.4 Find Sums of Infinite Geometric Series

703

For Your Notebook

KEY CONCEPT The Sum of an Infinite Geometric Series

The sum of an infinite geometric series with first term a1 and common ratio r is given by a 12r

1 S5}

provided r < 1. If r ≥ 1, the series has no sum.

EXAMPLE 2

Find sums of infinite geometric series

Find the sum of the infinite geometric series. `

a.

∑ 5(0.8)i 2 1

3 1 9 2 27 1 . . . b. 1 2 } } } 4 16 64

i51

Solution 3. b. For this series, a1 5 1 and r 5 2} 4 a1 1 4 S5} 5} 5} 12r 3 1 2 1 2} 2 7

a. For this series, a1 5 5 and r 5 0.8. a1 5 S5} 5} 5 25 12r 1 2 0.8

4

★

EXAMPLE 3

AVOID ERRORS

Standardized Test Practice

What is the sum of the infinite geometric series 1 2 3 1 9 2 27 1 . . . ?

If you substitute 1 for a1 and 23 for r in the a1

formula S 5 }, you

1 A }

12r

1 4

get an answer of S 5 } for the sum. However, this answer is not correct because the sum formula does not apply when r ≥ 1.

4 B }

4

C 3

3

D Does not exist

Solution 23 5 23. You know that a1 5 1 and a2 5 23. So, r 5 } 1

Because 23 ≥ 1, the sum does not exist. c The correct answer is D. A B C D

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

2 1 4 1 8 1 16 1 32 1 . . . . Find and graph the 1. Consider the series } } } } } 5 25 125 625 3125

partial sums Sn for n 5 1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases. Find the sum of the infinite geometric series, if it exists. `

2.

704

`

n21

∑ 1 2}12 2 n51

3.

∑ 3 1 }5 2 n51 4

Chapter 10 Sequences, Series, and the Binomial Theorem

n21

3 1 3 1 3 1... 4. 3 1 } } } 4 16 64

EXAMPLE 4

Use an infinite series as a model

PENDULUMS A pendulum that is released to swing freely travels 18 inches on the first swing. On each successive swing, the pendulum travels 80% of the distance of the previous swing. What is the total distance the pendulum swings?

Solution The total distance traveled by the pendulum is: d 5 18 1 18(0.8) 1 18(0.8)2 1 18(0.8) 3 1 . . . a 12r

1 5}

Write formula for sum.

18 5}

Substitute 18 for a1 and 0.8 for r.

5 90

Simplify.

1 2 0.8

c The pendulum travels a total distance of 90 inches, or 7.5 feet.

EXAMPLE 5

Write a repeating decimal as a fraction

Write 0.242424. . . as a fraction in lowest terms. 0.242424. . . 5 24(0.01) 1 24(0.01)2 1 24(0.01) 3 1 . . . a 12r

Write formula for sum.

5}

24(0.01) 1 2 0.01

Substitute 24(0.01) for a1 and 0.01 for r.

0.24 5}

Simplify.

24 5}

Write as a quotient of integers.

8 5}

Reduce fraction to lowest terms.

1 5}

0.99 99 33

8 as a fraction. c The repeating decimal 0.242424. . . is } 33

✓

GUIDED PRACTICE

for Examples 4 and 5

5. WHAT IF? In Example 4, suppose the pendulum travels 10 inches on its first

swing. What is the total distance the pendulum swings? Write the repeating decimal as a fraction in lowest terms. 6. 0.555. . .

7. 0.727272. . .

8. 0.131313. . .

10.4 Find Sums of Infinite Geometric Series

705

10.4

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS18 for Exs. 13, 27, and 39

★

MA.912.D.11.4

5 STANDARDIZED TEST PRACTICE Exs. 2, 32, 34, 39, 40, 41, and 42

SKILL PRACTICE 1. VOCABULARY Copy and complete: The sum Sn of the first n terms of an

infinite series is called a(n) ? .

`

2. ★ WRITING Explain how to tell whether the series

∑ a1r i 2 1 has a sum.

i51

EXAMPLE 1 on p. 703 for Exs. 3–6

EXAMPLES 2 and 3 on p. 704 for Exs. 7–23

PARTIAL SUMS For the given series, find and graph the partial sums Sn for

n 5 1, 2, 3, 4, and 5. Describe what happens to Sn as n increases. 1 1 1 1 1 1 1 1... 1 1} 3. } } } } 54 6 18 162 2

1 1 1 1 1 1 1 1... 2 1} 4. } } } } 3 6 12 24 3

12 1 36 1 108 1 324 1 . . . 5. 4 1 } } } } 5 25 125 625

5 1 25 1 125 1 625 1 . . . 1 1} 6. } } } } 4 4 4 4 4

FINDING SUMS Find the sum of the infinite geometric series, if it exists. `

7.

`

11.

`

∑ 8 1 }15 2n 2 1 n51 ∑ 2 1 }16 2

8.

9.

`

i21

12.

`

∑ 25 1 }25 2n 2 1 n51 `

∑ 9(4)k 2 1

16.

∑ 22 1 2}14 2

i21

k51

`

∑ 71 2}89 2k 2 1

14.

10 n 2 1 ∑ }1 1 2} 3 2 n51 2

18.

∑ }5 (3)n n50 6

k51

`

17.

`

∑ 1 2}37 2i

i50

19. ERROR ANALYSIS Describe and correct the

7. For this series, a1 5 1 and r 5 }

error in finding the sum of the infinite

2

`

geometric series

11 3 k 2 1 ∑} } 3 182

10.

`

13.

i51

k51

`

∑ }25 1 }53 2i 2 1

i51

k51

i51

15.

`

∑ 26 1 }32 2k 2 1

7 n 2 1. ∑ }2 1 n51 2

a1 1 5 1 5 22 S5} 5} } } 5 7 12r 5 2} 12} 2

2

FINDING SUMS Find the sum of the infinite geometric series, if it exists.

1 2 1 2 1 2 1 1... 20. 2} } } } 8 12 18 27

2 1 2 2 2 1... 2 2} 21. } } } 9 27 81 3

4 1 20 1 100 1 . . . 4 1} 22. } } } 9 27 81 15

5 1 25 1 125 1 . . . 23. 3 1 } } } 2 12 72

EXAMPLE 5

REWRITING DECIMALS Write the repeating decimal as a fraction in

on p. 705 for Exs. 24–32

lowest terms. 24. 0.222. . .

25. 0.444. . .

26. 0.161616. . .

27. 0.625625625. . .

28. 32.3232. . .

29. 130.130130. . .

30. 0.090909. . .

31. 0.2777. . .

32. ★ MULTIPLE CHOICE Which fraction is equal to the repeating decimal

18.1818. . . ? 2 A } 11

1836 B } 101

200 C }

33. REASONING Show that 0.999. . . is equal to 1.

706

Chapter 10 Sequences, Series, and the Binomial Theorem

11

181 D } 9

34. ★ OPEN-ENDED MATH Find two infinite geometric series whose sums are each 5. CHALLENGE Specify the values of x for which the given infinite geometric series

has a sum. Then find the sum in terms of x. 3 x 1 3 x2 1 3 x3 1 . . . 36. 6 1 } } } 2 8 32

35. 1 1 4x 1 16x 2 1 64x 3 1 . . .

PROBLEM SOLVING EXAMPLE 4 on p. 705 for Exs. 37–39

37. TIRE SWING A person is given one push on a tire swing and then allowed to

swing freely. On the first swing, the person travels a distance of 14 feet. On each successive swing, the person travels 80% of the distance of the previous swing. What is the total distance the person swings? For problem solving help, go to thinkcentral.com

38. BUSINESS A company had a profit of $350,000 in its first year. Since then, the

company’s profit has decreased by 12% per year. If this trend continues, what is an upper limit on the total profit the company can make over the course of its lifetime? Justify your answer using an infinite geometric series. For problem solving help, go to thinkcentral.com

39. ★ MULTIPLE CHOICE In 1994, the number of cassette tapes shipped in the

United States was 345 million. In each successive year, the number decreased by about 21.7%. What is the total number of cassettes that will ship in 1994 and after if this trend continues? A 420 million

B 440 million

C 615 million

D 1.59 billion

40. ★ SHORT RESPONSE Can the Greek hero Achilles, running at 20 feet

per second, ever catch up to a tortoise that runs 10 feet per second if the tortoise has a 20 foot head start? The Greek mathematician Zeno said no. He reasoned as follows:

When Achilles runs 20 feet, the tortoise will be in a new spot, 10 feet away.

Then, when Achilles gets to that spot, the tortoise will be 5 feet away.

Achilles will keep halving the distance but will never catch up to the tortoise.

In actuality, looking at the race as Zeno did, you can see that both the distances and the times Achilles required to traverse them form infinite geometric series. Using the table, show that both series have finite sums. Does Achilles catch up to the tortoise? Explain. Distance (ft)

20

10

5

2.5

1.25

0.625

...

Time (sec)

1

0.5

0.25

0.125

0.0625

0.03125

...

10.4 Find Sums of Infinite Geometric Series

707

41. ★ EXTENDED RESPONSE A student drops a rubber

8 ft

6 ft 1 6 ft

ball from a height of 8 feet. Each time the ball hits the ground, it bounces to 75% of its previous height.

4.5 ft 1 4.5 ft

a. How far does the ball travel between the first

and second bounces? between the second and third bounces?

3.375 ft 1 3.375 ft

2.531 ft 1 2.531 ft

b. Write an infinite series to model the total

distance traveled by the ball, excluding the distance traveled before the first bounce.

1

2

c. Find the total distance traveled by the ball,

3 4 Bounce number

including the distance traveled before the first bounce. d. Show that if the ball is dropped from a height of h feet, then the total

distance traveled by the ball (including the distance traveled before the first bounce) is 7h feet. 42. MULTI-STEP PROBLEM The Sierpinski triangle is a fractal created using

equilateral triangles. The process involves removing smaller triangles from larger triangles by joining the midpoints of the sides of the larger triangles as shown below. Assume that the initial triangle has an area of 1 square unit.

Stage 1

Stage 2

Stage 3

a. Copy and complete the table to find the area that is removed at each

stage in creating the Sierpinski triangle. Stage 1

Stage 2

Stage 3

Stage 4

Smallest triangle as a fraction of original triangle

}

1 4

}

1 16

}

1 64

?

Number of smallest triangles removed

1

3

?

?

Total area of triangles removed at the stage

}

1 4

?

?

?

b. Let an be the total area of the triangles that are removed in a given stage

n. Write a rule for an . `

c. Find

∑ an. What does your answer mean in the context of this n51

problem? n

43. REASONING Find the partial sums of the arithmetic series

∑ 0.02i

i51

for n 5 10, n 5 100, n 5 1000, and n 5 1,000,000. Does the infinite series `

∑ 0.02i have a sum? Can any infinite arithmetic series have a finite sum? i51 Explain your reasoning for both questions.

708

5 WORKED-OUT SOLUTIONS starting on p. WS1

★

5 STANDARDIZED TEST PRACTICE

5

44. CHALLENGE A Koch snowflake fractal is created by beginning with an

equilateral triangle with sides 1 unit long and successively adding new equilateral triangles on the middle third of each segment as shown below.

4UBHF


4UBHF


4UBHF
 }

s 2Ï 3 . For a. The area A of an equilateral triangle with side length s is A 5 } 4

each of the first 3 stages, find the number of new triangles added, the area of each new added triangle, and the total area added. Then write expressions that represent these quantities for the nth stage. b. What is the total area of the Koch snowflake? (Hint: Don’t forget the area

of the original triangle.)

FLORIDA SPIRAL REVIEW 3

}

x23 45. Which expression is equivalent to } ? (8.5) 21 1 } } x

x 23 B }

A x

MA.912.A.5.3

x23

2

x C }

x23

x D }

x22

x21

46. GRIDDED RESPONSE A car’s value has been decreasing by 14% per year since

it was new. After 6 years, the value is $8000. What was the original value of the car? Round your answer to the nearest hundred dollars. (7.2)

MA.912.A.8.7

READY TO GO ON?

QUIZ for Lessons 10.1–10.4

Find the sum of the series. (10.1) 5

4

1.

∑ 2i 3

2.

i51

∑ (k 2 1 3)

k51

6

3.

1 ∑} n52 n 2 1

Write a rule for the nth term an of the arithmetic or geometric sequence. Find a15, then find the sum of the first 15 terms of the sequence. 4. 1, 7, 13, 19, . . . (10.2)

7 , 5, . . . (10.2) 1 , 2, } 5. } 2 2

6. 5, 2, 21, 24, 27, . . . (10.2)

7. 2, 8, 32, 128, . . . (10.3)

4 , 8 , 16 , . . . (10.3) 8. 2, } } } 3 9 27

9. 23, 15, 275, 375, . . . (10.3)

Find the sum of the infinite geometric series, if it exists. (10.4) `

10.

∑ 2 1 }3 2 n51 7

`

n21

11.

∑ 4 1 2}56 2 n50

n

3 1 15 1 75 1 375 1 . . . 12. } } } } 4 8 16 32

13. PENDULUMS A pendulum that is released to swing freely travels 25 inches on

the first swing. On each successive swing, the pendulum travels 85% as far as the previous swing. What is the total distance the pendulum swings? (10.4)

EXTRA PRACTICE for Lesson 10.4, p. 787

ONLINE tonite thinkcentral.com 12.4 Find QUIZ Sums ofGoInfi Geometric Series

709

FLORIDA

Problem P Pr r roble m Solving Solvin Solv Sol Solvi S olving lv lving vin ng n g

CONNECTIONS CONNEC CONN C CO CON O ON ONNECTIONS ONN NN NN

Florida Test Practice Go to thinkcentral.com

Review Lessons 10.1–10.4 1. MULTI-STEP PROBLEM A ball is dropped from

5.

a height of 12 feet. With each bounce, the ball bounces to 70% of its previous height. a. Write an infinite series to model the total

distance traveled by the ball, excluding the distance traveled before the first bounce.

MULTI-STEP PROBLEM For 1997, the Gross Domestic Product (GDP) of Florida was about $391.5 billion. For the period from 1997 through 2007, the GDP of Florida increased at an average rate of about 6.50% per year. Port of Jacksonville

b. Find the total distance traveled by the ball,

including the distance traveled before the first bounce. 2. GRIDDED RESPONSE Pieces of chalk are

stacked in a pile. The bottom row has 15 pieces of chalk and the top row has 6 pieces of chalk. Each row has one fewer piece of chalk than the row below it. How many pieces of chalk are in the pile? 3. EXTENDED RESPONSE A target has rings that

are each 1 foot wide. a. Write a rule for the GDP a n (in billions

1 ft

of dollars) in terms of the year. Let n 5 1 represent 1997.

The 3 innermost rings of the target

b. What was the GDP of Florida in 2007? a. Write a rule for the area of the nth ring.

6. SHORT RESPONSE A builder is constructing a

staircase for a deck. At the foot of the staircase, there is a concrete slab that is 2 inches tall. Each stair is 7 inches tall. Write a rule for the height of the top of the nth stair. Find the height of the top of the 10th stair. Explain how you could modify the rule so that it gives the height of the bottom of the nth stair.

b. Use summation notation to write a series

that gives the total area of a target with n rings. c. Evaluate your expression from part (b) when

n 5 1, 2, 4, and 8. What effect does doubling the number of rings have on the area of the target?

7. SHORT RESPONSE The length l1 of the first loop

4. EXTENDED RESPONSE A scientist is studying

the radioactive decay of Platinum-197. The scientist starts with a 66 gram sample of Platinum-197 and measures the amount remaining every two hours. The amounts (in grams) recorded are 66, 33, 16.5, 8.25, . . . . a. Is this sequence arithmetic, geometric, or

neither? Explain how you know. b. Write a rule for the nth term of the sequence. c. Graph the sequence. Describe the curve on

which the points lie. d. After how many hours will the scientist first

measure an amount of Platinum-197 that is less than 1 gram?

710

Chapter 10 Sequences, Series, and the Binomial Theorem

of a spring is 16 inches. The length l2 of the second loop is 0.9 times the length of the first loop. The length l3 of the third loop is 0.9 times the length of the second loop, and so on. If the spring could have infinitely many loops, would its length be finite or infinite? Explain. If its length is finite, find the length.

FLORIDA

10.5 Before

Apply the Counting Principle and Permutations You counted the number of different ways to perform a task.

Now

Preparation for MA.912.A.4.12 Apply the Binomial Theorem.

Why

So you can find numbers of racing outcomes, as in Example 4.

Key Vocabulary • permutation • factorial

In many real-life problems, you want to count the number of ways to perform a task. One way to do this is to use a tree diagram.

EXAMPLE 1

Use a tree diagram

SNOWBOARDING A sporting goods store offers 3 types of snowboards (all-

mountain, freestyle, and carving) and 2 types of boots (soft and hybrid). How many choices does the store offer for snowboarding equipment? Solution Draw a tree diagram and count the number of branches. Soft

All-mountain board, soft boots

Hybrid

All-mountain board, hybrid boots

Soft

Freestyle board, soft boots

Hybrid

Freestyle board, hybrid boots

Soft

Carving board, soft boots

Hybrid

Carving board, hybrid boots

All-mountain

Freestyle

Carving

c The tree has 6 branches. So, there are 6 possible choices. FUNDAMENTAL COUNTING PRINCIPLE Another way to count the choices in

Example 1 is to use the fundamental counting principle. You have 3 choices for the board and 2 choices for the boots, so the total number of choices is 3 p 2 5 6.

For Your Notebook

KEY CONCEPT Fundamental Counting Principle

Two Events If one event can occur in m ways and another event can occur in

n ways, then the number of ways that both events can occur is m p n. Three or More Events The fundamental counting principle can be extended to three or more events. For example, if three events can occur in m, n, and p ways, then the number of ways that all three events can occur is m p n p p.

10.5 Apply the Counting Principle and Permutations

711

EXAMPLE 2

Use the fundamental counting principle

PHOTOGRAPHY You are framing a picture. The frames are available in 12

different styles. Each style is available in 55 different colors. You also want blue mat board, which is available in 11 different shades of blue. How many different ways can you frame the picture? Solution You can use the fundamental counting principle to find the total number of ways to frame the picture. Multiply the number of frame styles (12), the number of frame colors (55), and the number of mat boards (11). Number of ways 5 12 p 55 p 11 5 7260 c The number of different ways you can frame the picture is 7260.

EXAMPLE 3

Use the counting principle with repetition

LICENSE PLATES One configuration for

a Florida license plate has been 1 letter followed by 4 digits followed by 1 letter. (Assume all letters and digits are allowed.) a. How many different license plates are

possible if letters and digits can be repeated? b. How many different license plates are possible

if letters and digits cannot be repeated? Solution

AVOID ERRORS For a given situation, the number of choices without repetition is always less than the number of choices with repetition.

a. There are 26 choices for each letter and 10 choices for each digit. You can use

the fundamental counting principle to find the number of different plates. Number of plates 5 26 p 10 p 10 p 10 p 10 p 26 5 6,760,000

c With repetition, the number of different license plates is 6,760,000. b. If you cannot repeat letters there are still 26 choices for the first letter, but

then only 25 remaining choices for the second letter. Similarly, there are 10 choices for the first digit, 9 choices for the second digit, 8 choices for the third digit, and 7 choices for the fourth digit. You can use the fundamental counting principle to find the number of different plates. Number of plates 5 26 p 10 p 9 p 8 p 7 p 25 5 3,276,000 c Without repetition, the number of different license plates is 3,276,000.

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

1. SPORTING GOODS The store in Example 1 also offers 3 different types

of bicycles (mountain, racing, and BMX) and 3 different wheel sizes (20 in., 22 in., and 24 in.). How many bicycle choices does the store offer? 2. WHAT IF? In Example 3, how do the answers change for the configuration of

a New York license plate in which 3 letters are followed by 4 numbers?

712

Chapter 10 Sequences, Series, and the Binomial Theorem

PERMUTATIONS An ordering of n objects is a permutation of the objects. For instance, there are 6 permutations of the letters A, B, and C:

ABC

ACB

BAC

BCA

CAB

CBA

You can use the fundamental counting principle to find the number of permutations of A, B, and C. There are 3 choices for the first letter. After the first letter has been chosen, 2 choices remain for the second letter. Finally, after the first two letters have been chosen, there is only 1 choice remaining for the final letter. So, the number of permutations is 3 p 2 p 1 5 6. The expression 3 p 2 p 1 can also be written as 3!. The symbol ! is the factorial symbol, and 3! is read as “three factorial.” In general, n! is defined where n is a positive integer as follows:

DEFINE FACTORIALS Zero factorial is defined as 0! 5 1.

n! 5 n p (n 2 1) p (n 2 2) p . . . p 3 p 2 p 1 The number of permutations of n distinct objects is n!.

EXAMPLE 4

Find the number of permutations

OLYMPICS Ten teams are competing in the final round of the Olympic four-person bobsledding competition.

a. In how many different ways can the

bobsledding teams finish the competition? (Assume there are no ties.) b. In how many different ways can 3 of the

bobsledding teams finish first, second, and third to win the gold, silver, and bronze medals? Solution a. There are 10! different ways that the teams can finish the competition.

10! 5 10 p 9 p 8 p 7 p 6 p 5 p 4 p 3 p 2 p 1 5 3,628,800 b. Any of the 10 teams can finish first, then any of the remaining 9 teams can

finish second, and finally any of the remaining 8 teams can finish third. So, the number of ways that the teams can win the medals is: 10 p 9 p 8 5 720

✓

GUIDED PRACTICE

for Example 4

3. WHAT IF? In Example 4, how would the answers change if there were

12 bobsledding teams competing in the final round of the competition?

The answer to part (b) of Example 4 is called the number of permutations of 10 objects taken 3 at a time. It is denoted by 10P 3. Notice that this permutation can be computed using factorials: p9p8p7p6p5p4p3p2p1 10! 10! P 5 10 p 9 p 8 5 10 }}}}}}}}}}}}}} 5 } 5 }

10 3

7p6p5p4p3p2p1

7!

(10 2 3)!

This result is generalized at the top of the next page. 10.5 Apply the Counting Principle and Permutations

713

For Your Notebook

KEY CONCEPT Permutations of n Objects Taken r at a Time

The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr and is given by this formula: n! P 5}

n r

EXAMPLE 5

(n 2 r)!

Find permutations of n objects taken r at a time

MUSIC You are burning a demo CD for your band. Your band has 12 songs stored

on your computer. However, you want to put only 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD? Solution

EVALUATE PERMUTATIONS Most scientific and graphing calculators have a key or menu item for evaluating nPr .

✓

Find the number of permutations of 12 objects taken 4 at a time. 479,001,600 40,320

12! 12! 5 P 5 }}}} 5} } 5 11,880

12 4

(12 2 4)!

8!

c You can burn 4 of the 12 songs in 11,880 different orders.

GUIDED PRACTICE

for Example 5

Find the number of permutations. 4. 5P 3

5.

P

6.

4 1

P

7.

8 5

P

12 7

PERMUTATIONS WITH REPETITION If you consider the letters E and E to be

distinct, there are six permutations of the letters E, E, and Y: EEY EEY

EYE EYE

YEE YEE

However, if the two occurrences of E are considered interchangeable, then there are only three distinguishable permutations: EEY

EYE

YEE

Each of these permutations corresponds to two of the original six permutations because there are 2!, or 2, permutations of E and E. So, the number of 3! 5 6 5 3. permutations of E, E, and Y can be written as } } 2!

2

For Your Notebook

KEY CONCEPT Permutations with Repetition

The number of distinguishable permutations of n objects where one object is repeated s1 times, another is repeated s2 times, and so on, is: n! s1! p s2! p . . . p sk!

}}}}}}}

714

Chapter 10 Sequences, Series, and the Binomial Theorem

EXAMPLE 6

Find permutations with repetition

Find the number of distinguishable permutations of the letters in (a) MIAMI and (b) TALLAHASSEE. Solution a. MIAMI has 5 letters of which M and I are each repeated 2 times. So, the 5! 5 120 5 30. number of distinguishable permutations is }}} }}} 2p2 2! p 2! b. TALLAHASSEE has 11 letters of which A is repeated 3 times, and L, S,

and E are each repeated 2 times. So, the number of distinguishable 39,916,800 11! permutations is }}}}}} 5 }}}}} 5 831,600. 6p2p2p2 3! p 2! p 2! p 2!

✓

GUIDED PRACTICE

for Example 6

Find the number of distinguishable permutations of the letters in the word. 8. MALL

10.5

9. KAYAK

EXERCISES

10. CINCINNATI

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS19 for Exs. 13, 35, and 65

★

Preparation for MA.912.A.4.12

5 STANDARDIZED TEST PRACTICE Exs. 2, 17, 42, 55, 57, and 68

SKILL PRACTICE 1. VOCABULARY What is a permutation of n objects? 2. ★ WRITING Simplify the formula for nPr when r 5 0. Explain why this result

makes sense.

EXAMPLE 1

TREE DIAGRAMS An object has an attribute from each list. Make a tree diagram

on p. 711 for Exs. 3–6

that shows the number of different objects that can be created. 3.

T-Shirts

4.

Toast

Size: M, L, XL

Bread: white, wheat

Type: long-sleeved, short-sleeved

Spread: jam, margarine

5.

Meal

6.

Furniture

Entrée: chicken, fish, pasta

Wood: cherry, mahogany, oak, pine

Side: corn, green beans, potato

Finish: stained, painted, unfinished

EXAMPLE 2

FUNDAMENTAL COUNTING PRINCIPLE Each event can occur in the given number

on p. 712 for Exs. 7–10

of ways. Find the number of ways all of the events can occur. 7. Event A: 2 ways; Event B: 4 ways

8. Event A: 5 ways; Event B: 2 ways

9. Event A: 4 ways; Event B: 3 ways;

10. Event A: 3 ways; Event B: 6 ways;

Event C: 5 ways

Event C: 5 ways; Event D: 2 ways

10.5 Apply the Counting Principle and Permutations

715

EXAMPLE 3

LICENSE PLATES For the given configuration, determine how many different

on p. 712 for Exs. 11–17

license plates are possible if (a) digits and letters can be repeated, and (b) digits and letters cannot be repeated. 11. 4 letters followed by 3 digits

12. 2 letters followed by 5 digits

13. 4 letters followed by 2 digits

14. 5 digits followed by 3 letters

15. 1 digit followed by 5 letters

16. 6 letters

17. ★ MULTIPLE CHOICE How many different license plates with 2 letters

followed by 4 digits are possible if digits and letters cannot be repeated? A 3,276,000 EXAMPLES 4 and 5 on pp. 713–714 for Exs. 18–41

B 6,760,000

C 32,292,000

D 45,697,600

FACTORIALS Evaluate the expression.

18. 7!

19. 11!

20. 1!

21. 8!

22. 4!

23. 0!

24. 12!

25. 6!

26. 3! p 4!

27. 3(4!)

8! 28. } (8 2 5)!

9! 29. } 4! p 4!

PERMUTATIONS Find the number of permutations.

30.

4 4

P

31.

6 2

P

32.

10 1

P

33.

8 7

P

34.

7 4

P

35.

9 2

P

36.

13 8

P

37.

7 7

38.

5 0

P

39.

9 4

P

40.

11 4

P

41.

15 0

P

P

42. ★ SHORT RESPONSE Let n be a positive integer. Find the number of

permutations of n objects taken n 2 1 at a time. Compare your answer with the number of permutations of all n objects. Does this make sense? Explain. EXAMPLE 6

PERMUTATIONS WITH REPETITION Find the number of distinguishable

on p. 715 for Exs. 43–55

permutations of the letters in the word. 43. OFF

44. TREE

45. SKILL

46. YELLOW

47. GRAVEL

48. PANAMA

49. ARKANSAS

50. FACTORIAL

51. MAGNETIC

52. HONOLULU

53. CLEVELAND

54. MISSISSIPPI

55. ★ MULTIPLE CHOICE What is the number of distinguishable permutations

of the letters in the word HAWAII? A 24

B 180

C 360

56. ERROR ANALYSIS In bingo, balls labeled from 1 to 75 are

drawn from a container without being replaced. Describe and correct the error in finding the number of ways the first 4 numbers can be chosen for a game of bingo.

D 720

75 p 75 p 75 p 75 5 31,640,625

57. ★ SHORT RESPONSE Explain how the fundamental counting principle

can be used to justify the formula for the number of permutations of n distinct objects. SOLVING EQUATIONS Solve for n.

58.

P 5 8(nP 3)

n 4

59.

P 5 5(nP5)

n 6

60.

P 5 9(nP4)

n 5

61. CHALLENGE Find the number of distinguishable permutations of 6 letters

that are chosen from the letters in the word MANATEE.

716

5 WORKED-OUT SOLUTIONS

starting onMethods p. WS1 and Probability Chapter 10 Counting

★

5 STANDARDIZED TEST PRACTICE

PROBLEM SOLVING EXAMPLE 2 on p. 712 for Exs. 62–63

62. CLASS RINGS You want to purchase a class ring. The ring can be made

from 3 different metals. You can choose from 6 different side designs and 12 different stones. How many different class rings are possible? Metal

Side Design

Stone

Auralite

Academics

Literature

Gold

Art

Music

Silver

Athletics

Technology

For problem solving help, go to thinkcentral.com

63. ENVIRONMENT Since 1990, the Goldman Environmental Prize has been

awarded annually to 6 grassroots environmentalists, one from each of 6 regions. The regions consist of 52 countries in Africa, 47 in Europe, 45 in Asia, 36 in island nations, 19 in South and Central America, and 3 in North America. How many different sets of 6 countries can be represented by the prize winners in a given year? For problem solving help, go to thinkcentral.com

EXAMPLES 4, 5, and 6 on pp. 713–715 for Exs. 64–66

64. PHOTOGRAPHY A photographer lines up the 15 members of a family in a

single line in order to take a photograph. How many different ways can the photographer arrange the family members for the picture? 65. SCHOOL CLUBS A Spanish club is electing a president, vice president, and

secretary. The club has 9 members who are eligible for these offices. How many different ways can the 3 offices be filled? 66. MUSIC The window of a music store has 8 stands in fixed positions where

instruments can be displayed. In how many ways can 3 identical guitars, 2 identical keyboards, and 3 identical violins be displayed? 67. MULTI-STEP PROBLEM You are designing an entertainment center. You want

to include three audio components and three video components. a. You want one of each audio component

listed at the right. How many selections of audio components are possible? b. You want one of each video component

listed at the right. How many selections of video components are possible? c. How many selections of all six audio

Entertainment Center Audio Components

Video Components

5 receivers

7 TV sets

8 CD players

9 DVD players

6 speakers

4 game systems

and video components are possible? 68. ★ EXTENDED RESPONSE To keep computer files secure, many programs

require the user to enter a password. The shortest allowable passwords are typically 6 characters long and can contain both letters and digits. a. Calculate How many 6-character passwords are possible if characters

can be repeated? b. Calculate How many 6-character passwords are possible if characters

cannot be repeated? c. Draw Conclusions Which type of password is more secure? Explain. 10.5 Apply the Counting Principle and Permutations

717

69. CLOTHING DISPLAY An employee at a clothing store is creating a display. The

display has 3 different mannequins. Each mannequin is to wear a different sweater and a different skirt. How many different displays can be created?

Just Arrived......

Sweaters On Sale......

7 NEW SKIRT STYLES

8 DIFFERENT COLORS

"MHFCSB Go to thinkcentral.com 70. CROSS COUNTRY Three schools are competing in a cross country race.

School A has 6 runners, school B has 5 runners, and school C has 4 runners. For scoring purposes, the finishing order of the race only considers the school of each runner. How many different finishing orders are there for the 15 runners? 71. CHALLENGE You have learned that n! represents the number of ways that

n objects can be placed in a linear order, where it matters which object is placed first. Now consider circular permutations in which objects are placed in a circle, so that it does not matter which object is placed first. a. Suppose you are seating 5 people at a circular

table. How many different ways can you arrange the people around the table?

!

# "

%

b. Find a formula for the number of permutations

of n objects placed in clockwise order around a circle when only the relative order of the objects matters. Explain how you derived your formula.

$

#

72. GRIDDED RESPONSE In 1995, the average U.S. public college tuition was

$2057. From 1995 through 2002, this amount increased by about 6% per year. What is the value to the nearest dollar of a 7 in the sequence of terms a n that gives the average tuition for the period where n 5 1 represents 1995? (10.3) 73. Identical rectangular tables fill a triangular area

of a banquet hall. A single table occupies the narrowest end of the triangle. Next to it are two tables placed end-to-end as shown. The next row of tables has three tables placed end-to-end, and so on. In all, there are 13 rows of tables. What is the total number of people who can be seated? (10.2) MA.912.D.11.1

A 351

B 390

C 416

D 780

74. GRIDDED RESPONSE A geometric series has a first term of 2125 and a

MA.912.D.11.4

718

common ratio of 20.8. What is S 6, the sixth partial sum of the series? (10.4)

EXTRA PRACTICE for Lesson 10.5, p. 787

!

%

The two arrangements shown represent the same permutation.

FLORIDA SPIRAL REVIEW

MA.912.D.11.3

$

"

ONLINE QUIZ Go to thinkcentral.com

FLORIDA

10.6 Before

Use Combinations and the Binomial Theorem You applied the counting principle and permutations.

Now

MA.912.A.4.12 Apply the Binomial Theorem.

Why

So you can find combinations of plays, as in Example 2.

Key Vocabulary • combination • Pascal’s triangle • binomial theorem

With permutations, the order of the objects is important. If you buy 5 DVDs, however, the order in which you choose them is not important. A combination is a selection of r objects from a group of n objects where order is not important.

For Your Notebook

KEY CONCEPT Combinations of n Objects Taken r at a Time

The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr and is given by this formula: n! C 5 }}}}}

n r

(n 2 r)! p r!

Each combination of r objects from n objects has r! possible orders. This means that nC r p r! 5 nP r. Dividing both sides of this equation by r! gives nC r 5 nP r 4 r!.

EXAMPLE 1

Find combinations

CARDS A standard deck of 52 playing cards has

4 suits with 13 different cards in each suit. a. If the order in which the cards are dealt is

not important, how many different 5-card hands are possible? b. In how many 5-card hands are all 5 cards of

the same color? Solution a. The number of ways to choose 5 cards

from a deck of 52 cards is:

Standard 52-Card Deck K Q J 10 9 8 7 6 5 4 3 2 A

K Q J 10 9 8 7 6 5 4 3 2 A

K Q J 10 9 8 7 6 5 4 3 2 A

K Q J 10 9 8 7 6 5 4 3 2 A

52! 5 52 p 51 p 50 p 49 p 48 p 47! 5 2,598,960 C 5 }}}} }}}}}}}}}}}

52 5

47! p 5!

47! p 5!

b. For all 5 cards to be the same color, you must choose 1 of the 2 colors

and then 5 of the 26 cards in that color. The number of possible hands is: 2! p 26! 5 2 p 26 p 25 p 24 p 23 p 22 p 21! 5 131,560 C p 26C5 5 }}} }}}} }}} }}}}}}}}}}}

2 1

1! p 1!

21! p 5!

1p1

21! p 5!

10.6 Use Combinations and the Binomial Theorem

719

MULTIPLE EVENTS When finding the number of ways both an event A and an

event B can occur, you need to multiply, as in part (b) of Example 1. When finding the number of ways that event A or event B can occur, you add instead.

EXAMPLE 2

Decide to multiply or add combinations

THEATER William Shakespeare wrote 38 plays that can be divided into three

genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies. a. How many different sets of exactly 2 comedies and 1 tragedy can you read? b. How many different sets of at most 3 plays can you read? AVOID ERRORS

Solution

When finding the number of ways to select at most n objects, be sure to include the possibility of selecting 0 objects.

a. You can choose 2 of the 18 comedies and 1 of the 10 tragedies. So, the

number of possible sets of plays is: 18! p 10! 5 18 p 17 p 16! p 10 p 9! 5 153 p 10 5 1530 C p 10C1 5 }}}} }}} }}}}}} }}} 16! p 2! 9! p 1! 16! p 2 p 1 9! p 1

18 2

b. You can read 0, 1, 2, or 3 plays. Because there are 38 plays that can be

chosen, the number of possible sets of plays is: C 1 38C1 1 38C2 1 38C 3 5 1 1 38 1 703 1 8436 5 9178

38 0

SUBTRACTING POSSIBILITIES Counting problems that involve phrases like “at least” or “at most” are sometimes easier to solve by subtracting possibilities you do not want from the total number of possibilities.

EXAMPLE 3

Solve a multi-step problem

BASKETBALL During the school year, the girl’s basketball team is scheduled

to play 12 home games. You want to attend at least 3 of the games. How many different combinations of games can you attend? Solution Of the 12 home games, you want to attend 3 games, or 4 games, or 5 games, and so on. So, the number of combinations of games you can attend is: C 1 12C4 1 12C5 1 . . . 1 12C12

12 3

Instead of adding these combinations, use the following reasoning. For each of the 12 games, you can choose to attend or not attend the game, so there are 212 total combinations. If you attend at least 3 games, you do not attend only a total of 0, 1, or 2 games. So, the number of ways you can attend at least 3 games is: 212 2 (12C 0 1 12C1 1 12C2 ) 5 4096 2 (1 1 12 1 66) 5 4017

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

Find the number of combinations. 1. 8C3

2.

C

10 6

3. 7C2

4.

C

14 5

5. WHAT IF? In Example 2, how many different sets of exactly 3 tragedies and

2 histories can you read?

720

Chapter 10 Sequences, Series, and the Binomial Theorem

PASCAL’S TRIANGLE If you arrange the values of nCr in a triangular pattern in which each row corresponds to a value of n, you get what is called Pascal’s triangle. Pascal’s triangle is named after the French mathematician Blaise Pascal (1623−1662).

For Your Notebook

KEY CONCEPT Pascal’s Triangle

Pascal’s triangle is shown below with its entries represented by combinations and with its entries represented by numbers. The first and last numbers in each row are 1. Every number other than 1 is the sum of the closest two numbers in the row directly above it. Pascal’s triangle as combinations

Pascal’s triangle as numbers

C

1

n 5 0 (0th row)

0 0

n 5 1 (1st row)

C

n 5 2 (2nd row)

C

n 5 3 (3rd row)

C

C

n 5 5 (5th row)

EXAMPLE 4

C

4 0

C

5 0

C

C

3 3

C

4 2

5 3

1

C

4 3

C

5 2

1

C

3 2

C

4 1

5 1

2 2

C

3 1

1

C

2 1

C

3 0

n 5 4 (4th row)

1 1

C

2 0

1

C

1 0

4 4

C

5 4

1

C

5 5

2 3

4 5

1 1 3 6

1 4

10 10

1 5

1

Use Pascal’s triangle

SCHOOL CLUBS The 6 members of a Model UN club must choose 2 representatives

to attend a state convention. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives. Solution Because you need to find 6C2, write the 6th row of Pascal’s triangle by adding numbers from the previous row. n 5 5 (5th row) n 5 6 (6th row)

1 1 C 6 0

5 6 C 6 1

10 15 C 6 2

20 C 6 3

10 15 C 6 4

5

1 6 C 6 5

1 C 6 6

c The value of 6C2 is the third number in the 6th row of Pascal’s triangle, as shown above. Therefore, 6C2 5 15. There are 15 combinations of representatives for the convention.

✓

GUIDED PRACTICE

for Example 4

6. WHAT IF? In Example 4, use Pascal’s triangle to find the number of

combinations of 2 members that can be chosen if the Model UN club has 7 members.

10.6 Use Combinations and the Binomial Theorem

721

BINOMIAL EXPANSIONS There is an important relationship between powers of binomials and combinations. The numbers in Pascal’s triangle can be used to find coefficients in binomial expansions. For example, the coefficients in the expansion of (a 1 b)4 are the numbers of combinations in the row of Pascal’s triangle for n 5 4:

(a 1 b)4 5 1a 4 1 4a 3b 1 6a2b2 1 4ab3 1 1b4 C C C C C 4 0 4 1 4 2 4 3 4 4 This result is generalized in the binomial theorem.

For Your Notebook

KEY CONCEPT Binomial Theorem

For any positive integer n, the binomial expansion of (a 1 b) n is: (a 1 b) n 5 nC 0 anb 0 1 nC1an 2 1b1 1 nC2an 2 2b2 1 . . . 1 nCna 0bn Notice that each term in the expansion of (a 1 b) n has the form nCr an 2 rbr where r is an integer from 0 to n.

EXAMPLE 5

Expand a power of a binomial sum

Use the binomial theorem to write the binomial expansion. (x2 1 y) 3 5 3C 0(x2)3y 0 1 3C1(x2)2y1 1 3C2(x2)1y 2 1 3C3(x2)0y 3 5 (1)(x6)(1) 1 (3)(x4)(y) 1 (3)(x2)( y 2) 1 (1)(1)( y 3) 5 x6 1 3x4y 1 3x2y 2 1 y 3

POWERS OF BINOMIAL DIFFERENCES To expand a power of a binomial difference,

you can rewrite the binomial as a sum. The resulting expansion will have terms whose signs alternate between 1 and 2.

EXAMPLE 6

AVOID ERRORS When a binomial has a term or terms with a coefficient other than 1, the coefficients of the binomial expansion are not the same as the corresponding row of Pascal’s triangle.

✓

Expand a power of a binomial difference

Use the binomial theorem to write the binomial expansion. (a 2 2b)4 5 [a 1 (22b)]4 5 4C 0 a 4(22b) 0 1 4C1a3 (22b)1 1 4C2a2 (22b)2 1 4C3a1(22b) 3 1 4C4a 0 (22b)4 5 (1)(a 4)(1) 1 (4)(a3)(22b) 1 (6)(a2)(4b2) 1 (4)(a)(28b3) 1 (1)(1)(16b4) 5 a 4 2 8a 3b 1 24a2b2 2 32ab3 1 16b4

GUIDED PRACTICE

for Examples 5 and 6

Use the binomial theorem to write the binomial expansion. 7. (x 1 3) 5

722

8. (a 1 2b)4

Chapter 10 Sequences, Series, and the Binomial Theorem

9. (2p 2 q)4

10. (5 2 2y) 3

EXAMPLE 7

Find a coefficient in an expansion

Find the coefficient of x 4 in the expansion of (3x 1 2)10. Solution From the binomial theorem, you know the following: (3x 1 2)10 5 10C 0 (3x)10 (2) 0 1 10C1(3x) 9 (2)1 1 . . . 1 10C10 (3x) 0 (2)10 Each term in the expansion has the form 10Cr (3x)10 2 r (2) r. The term containing x4 occurs when r 5 6: C (3x)4(2) 6 5 (210)(81x4)(64) 5 1,088,640x4

10 6

c The coefficient of x4 is 1,088,640.

✓

GUIDED PRACTICE

for Example 7

11. Find the coefficient of x5 in the expansion of (x 2 3)7. 12. Find the coefficient of x 3 in the expansion of (2x 1 5) 8.

10.6

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS19 for Exs. 17, 29, and 49

★

MA.912.A.4.12

5 STANDARDIZED TEST PRACTICE Exs. 2, 38, 39, and 52

SKILL PRACTICE 1. VOCABULARY Copy and complete: The binomial expansion of (a 1 b) n is

given by the ? . 2. ★ WRITING Explain the difference between permutations and combinations. EXAMPLES 1, 2, and 3 on pp. 719–720 for Exs. 3–18

COMBINATIONS Find the number of combinations.

C

C

5.

9 6

6.

C

9. 7C 5

10.

3.

5 2

4.

10 3

7.

11 11

C

8.

12 4

C

C

8 2

C

14 6

ERROR ANALYSIS Describe and correct the error in finding the number of

combinations. 11.

12.

6! 720 5 30 C 5 }}}} 5 }} 6 2 24 (6 2 2)!

40,320 6

8! 5 C 5 }} }}}} 5 6720

8 3

3!

CARD HANDS Find the number of possible 5-card hands that contain the cards

specified. The cards are taken from a standard 52-card deck. 13. 5 face cards (kings, queens, or jacks)

14. 4 kings and 1 other card

15. 1 ace and 4 cards that are not aces

16. 5 hearts or 5 diamonds

17. At most 1 queen

18. At least 1 spade

10.6 Use Combinations and the Binomial Theorem

723

EXAMPLE 4

19. USING PATTERNS Copy Pascal’s triangle on page 721 and add rows for

n 5 6, 7, 8, 9, and 10.

on p. 721 for Exs. 19–23

PASCAL’S TRIANGLE Use the rows of Pascal’s triangle from Exercise 19 to write

the binomial expansion. 20. (x 1 3) 6 EXAMPLES 5 and 6 on p. 722 for Exs. 24–31

22. (a 1 b2)8

21. (y 2 3z)10

23. (2s 2 t 4)7

BINOMIAL THEOREM Use the binomial theorem to write the binomial expansion.

24. (x 1 2) 3

25. (c 2 4) 5

26. (a 1 3b)4

27. (4p 2 q) 6

28. (w 3 2 3)4

29. (2s4 1 5) 5

30. (3u 1 v 2)6

31. (x 3 2 y 2)4

EXAMPLE 7

32. Find the coefficient of x5 in the expansion of (x 2 2)10.

on p. 723 for Exs. 32–35

33. Find the coefficient of x3 in the expansion of (3x 1 2) 5. 34. Find the coefficient of x6 in the expansion of (x2 2 3) 8. 35. Find the coefficient of x4 in the expansion of (x 2 3)7. REASONING In Exercises 36 and 37, decide whether the problem requires

combinations or permutations to find the answer. Then solve the problem. 36. NEWSPAPER Your school newspaper has an editor-in-chief and an assistant

editor-in-chief. The staff of the newspaper has 12 students. In how many ways can students be chosen for these two positions? 37. STUDENT COUNCIL Five representatives from a senior class of 280 students

are to be chosen for the student council. In how many ways can students be chosen to represent the senior class on the student council? 38. ★ MULTIPLE CHOICE A relay race has a team of 4 runners who run different

parts of the race. There are 20 students on your track squad. In how many ways can the coach select students to compete on the relay team? A 4845

B 40,000

C 116,280

D 160,000

39. ★ SHORT RESPONSE Write the sequence 4C r for r 5 0, 1, 2, 3, 4. Describe the

symmetry. Explain why the symmetry exists.

PASCAL’S TRIANGLE In Exercises 40 and 41, use the diagrams shown.

40. What is the sum of the numbers in

41. Write the sequence formed by the sums

each of rows 0–4 of Pascal’s triangle? Use sigma notation and combinations notation to write a formula for the sum of the numbers in row n.

of the numbers along each diagonal of Pascal’s triangle shown. Describe the pattern that you see. Then write the next three terms of the sequence.

Row 0 Row 1 Row 2 Row 3 Row 4

CHALLENGE Verify the identity. Justify each of your steps.

724

C 51

43.

n n

C 5 nP1

46.

n r

42.

n 0

45.

n 1

5 WORKED-OUT SOLUTIONS

Chapter 10 Counting Methods starting on p. WS1 and Probability

C p rCm 5 nCm p n 2 mCr 2 m

C 51

44.

n r

C 5 nCn 2 r

47.

n11 r

★

5 STANDARDIZED TEST PRACTICE

C 5 nCr 1 nCr 2 1

PROBLEM SOLVING EXAMPLES 1, 2, and 3 on pp. 719–720 for Exs. 48–50

48. MUSIC You want to purchase 3 CDs from an

online collection that contains the types of music shown at the right. You want each CD to contain a different type of music such that 2 CDs are different types of contemporary music and 1 CD is a type of classical music. How many different sets of music types can you choose?

#$S

#ONTEMPORARY

For problem solving help, go to thinkcentral.com

"LUES #OUNTRY *AZZ 2AP 2OCK2OLL

#LASSICAL /PERA #ONCERTO 3YMPHONY

49. FLOWERS You are buying a bouquet. The florist has 18 types of flowers that

you can use to make the bouquet. You want to use exactly 3 types of flowers. How many different combinations of flower types can you use in your bouquet? For problem solving help, go to thinkcentral.com

50. ARCADE GAMES An arcade has 20 different arcade games. You want to play

at least 14 of them. How many different combinations of arcade games can you play? 51. MULTI-STEP PROBLEM A televised singing competition picks a winner from

20 original contestants over the course of five episodes. During each of the first, second, and third episodes, 5 singers are eliminated by the end of the episode. The fourth episode eliminates 2 more singers, and the winner is selected at the end of the fifth episode. a. How many combinations of 5 singers out of the original 20 can be

eliminated during the first episode? b. How many combinations of 5 singers out of the 15 singers who started

the second episode can be eliminated during the second episode? c. How many combinations of singers can be eliminated during the third

episode? during the fourth episode? during the fifth episode? d. Find the total number of ways in which the 20 original contestants can be

eliminated to produce a winner. 52. ★ EXTENDED RESPONSE A group of 15 high school students is volunteering

at a local fire station. Of these students, 5 will be assigned to wash fire trucks, 7 will be assigned to repaint the station’s interior, and 3 will be assigned to do maintenance on the station’s exterior. a. Calculate One way to count the number of possible job

assignments is to find the number of permutations of 5 W’s (for “wash”), 7 R’s (for “repainting”), and 3 M’s (for “maintenance”). Use this method to write the number of possible job assignments first as an expression involving factorials and then as a number. b. Calculate Another way to count the number of possible

job assignments is to first choose the 5 W’s, then choose the 7 R’s, and then choose the 3 M’s. Use this method to write the number of possible job assignments first as an expression involving factorials and then as a number. c. Analyze Compare your results from parts (a) and (b).

Explain why they make sense.

Volunteers in Aniak, Alaska

10.6 Use Combinations and the Binomial Theorem

725

diagonal

53. CHALLENGE A polygon is convex if no line that contains

a side of the polygon contains a point in the interior of the polygon. Consider a convex polygon with n sides. a. Use the combinations formula to write an expression

for the number of line segments that join pairs of vertices on an n-sided polygon.

vertex

b. Use your result from part (a) to write a formula for the

number of diagonals of an n-sided convex polygon.

FLORIDA SPIRAL REVIEW 54. You go online to pick seats for a concert. Several

F 1 3 5 7 9 11 13 15 17

rows of the seating chart in the section where you wish to sit are shown. The pattern of adding one seat per row after row B continues through the last row of the section, row M. Which expression gives the total number of seats in the section? (10.1) 12

A

∑ (3 1 (4 1 i))

B 31

∑ (4 1 i)

C 1 3 5 7 9 11 B 1 3 5 7 9 A 1 3 5

∑ (5 1 i)

i51 13

12

C 31

D 1 3 5 7 9 11 13

12

i51

MA.912.D.11.2

E 1 3 5 7 9 11 13 15

D

i51

∑ (3 1 i)

i51

55. GRIDDED RESPONSE The intensity I of a sound (in watts per square meter)

varies inversely with the square of the distance d (in meters) from the source of the sound. At a distance of 1 meter from the stage, the sound intensity at a rock concert is 10 watts per square meter. What is the intensity in watts per square meter of the sound you hear if you are 20 meters from the stage? (8.1)

MA.912.A.2.12

READY TO GO ON?

QUIZ for Lessons 10.5–10.6

For the given license plate configuration, find how many plates are possible if letters and digits (a) can be repeated and (b) cannot be repeated. (10.5) 1. 2 letters followed by 3 digits

2. 3 digits followed by 3 letters

Find the number of distinguishable permutations of the letters in the word. (10.5) 3. AWAY

4. IDAHO

5. LETTER

6. TENNESSEE

Find the number of combinations. (10.6) 7.

C

8 6

8. 7C4

9.

C

9 0

10.

C

12 11

Use the binomial theorem to write the binomial expansion. (10.6) 11. (x 1 5) 5

12. (2s 2 3) 6

13. (3u 1 v)4

14. (2x3 2 3y) 5

15. Find the coefficient of x 3 in the expansion of (x 1 2) 9. (10.6) 16. MENU CHOICES A pizza parlor runs a special where you can buy a large

pizza with 1 cheese, 1 vegetable, and 2 meats for $12. You have a choice of 5 cheeses, 10 vegetables, and 6 meats. How many different variations of the pizza special are possible? (10.6)

726

EXTRA PRACTICE for Lesson 10.6, p. 787

ONLINE QUIZ Go to thinkcentral.com

FLORIDA A

Problem P Pr r roble m Solvin Solving Solv Sol Solvi S olving lv lving vin ng n g

CONNECTIONS CONNEC CONN C CO CON O ONN ONNECTIONS ON NN NN

Florida Test Practice Go to thinkcentral.com

Review Lessons 10.5–10.6 1. MULTI-STEP PROBLEM Five people walk into a

movie theater and look for empty seats. a. Find the number of ways the people can be

seated if there are 5 empty seats. b. Find the number of ways the people can be

seated if there are 8 empty seats.

6.

MULTI-STEP PROBLEM Florida has 118 species of wild orchids, almost half the species in all of North America. Your science fair project about Florida’s wild orchids will include a display showing photos of 10 of the 118 species. Butterfly Orchid

c. Generalize your results from parts (a)

and (b) by writing an expression involving factorials for the number of ways the people can be seated if there are n empty seats. d. What is the minimum value of n such that

there are at least 1 million ways the people can be seated? 2. SHORT RESPONSE Write expressions for the numbers of distinguishable permutations of the letters in the place names OKEFENOKEE, OKEECHOBEE, and WACCASASSA. Then, without evaluating the expressions, order the names from the least to the greatest number of distinguishable permutations. Explain how you determined the order.

a. About how many possible combinations of

10 species can you choose from? b. All 43 orchid species found near Naples,

3. GRIDDED RESPONSE You want to make a fruit

smoothie using 3 of the fruits listed. How many different fruit smoothies can you make?

Florida, are on the endangered species list. How many possible groups of 10 species can you choose from only these 43? c. Once you have chosen 10 photos, in how

• O r an g e • Banan a • S tr aw be r • Pineap r y ple • Cant e loupe • Water melon • Kiwi • Peach

many ways can you arrange them in a row? 7. SHORT RESPONSE You must take 18 elective

courses to meet your graduation requirements for college. There are 30 courses that you are interested in. Does finding the number of possible course selections involve permutations or combinations? Explain. How many different course selections are possible? 8. OPEN-ENDED Give a real-life problem for which

4. GRIDDED RESPONSE In a high school fashion

show, how many ways can 1 freshman, 2 sophomores, 2 juniors, and 3 seniors line up in front of the judges if the contestants in the same class are considered identical?

the answer is the product of two combinations. Show how to find the answer. 9. EXTENDED RESPONSE Consider the binomial

expansions of (a 1 b) n and (a 2 b) n. a. Write the expansions of each binomial for

n 5 2, 3, and 4. 5. GRIDDED RESPONSE An ice cream shop offers

a choice of 31 flavors. How many different ice cream cones can be made with three scoops of ice cream if each scoop is a different flavor and the order of the scoops is not important?

b. Using your work in part (a), find the sum

(a 1 b) n 1 (a 2 b) n for n 5 2, 3, and 4. c. Explain how you can use your work and

Pascal’s triangle to find (a 1 b) 5 1 (a 2 b) 5. Problem Solving Connections

727

10

FLORIDA

RE READING REA RE EA EADING A & WRITING MATH

STUDY STRATEGY: WRITE PROCEDURES LA.910.4.2.1 The student will write in a variety of informational/ expository forms, including a variety of technical documents (e.g., how-to manuals, procedures, assembly directions).

The Sierpinski carpet is a fractal that is created using squares. The figure below shows the fi rst three stages of the fractal.

Stage 1

Stage 2

Stage 3

PRACTICE 1. Write a brief procedure for creating the Sierpinski carpet. Be sure

that someone could create the Sierpinski carpet using only the directions you provide. 2. Let an be the number of squares removed at the nth stage. Find a

rule for an . Then explain how to find the total number of squares removed through stage 8. 3. Assume that each side of the original square is one unit long. Let bn

be the remaining area of the original square after the nth stage. Find a rule for bn . Then explain how to find the remaining area of the original square after stage 12.

4. Describe what happens to the remaining area of the original square

as you create more and more stages of the fractal. For Exercises 5–8, refer to the diagram of Pascal’s triangle shown below. 5. Write a procedure for generating

Pascal’s triangle, beginning with rows 0 and 1.

1 1

6. Write a procedure for using Pascal’s

1

triangle to find nCr for given values of n and r.

1

7. Write a procedure for using Pascal’s

triangle to find the total number of possible combinations when you choose no more than 5 out of 8 items.

1 1

2 3

4 5

1 1 3 6

10 10

1 4

1 5

1

8. Write a real-world problem that can be solved using the procedure

you wrote in Exercise 7.

728

Chapter 12 Reading and Sequences Writing Math and Series

10

CHAPTER CHAPTER CH H SUMMARY

FLORIDA

KEY IDEAS For Your Notebook Key Idea 1

Analyze Sequences Arithmetic Sequence

Geometric Sequence

an

First term: a1

Common difference: d

Common ratio: r Graph is exponential.

1

n

1

an 5 a1r n 2 1

First term: a1

Graph is linear.

1

Key Idea 2

an

an 5 a1 1 (n 2 1)d

n

1

Find Sums of Series Arithmetic Series

Geometric Series

Infinite Geometric Series

Sum of the first n terms:

Sum of the series:

1 2 rn Sn 5 a1 } , r Þ 1 12r

S 5 }, r < 1

Example:

Example:

Example:

4 1 9 1 14 1 19 1 24

3 1 6 1 12 1 24

5 1 1 1 0.2 1 0.04 1 . . .

4 1 24 S5 5 5 } 5 70

122 S4 5 3 } 5 45

Sum of the first n terms:

1

a1 1 an

Sn 5 n }

1

2

2

2

1

a1

2

12r

4

5 1 2 0.2

1 122 2

2

S 5 } 5 6.25

Other common sum formulas: n

∑15n

i51

Key Idea 3

n

n

n(n 1 1) ∑i5 } 2

n(n 1 1)(2n 1 1) ∑ i2 5 } 6

i51

i51

Using Permutations and Combinations PERMUTATIONS Order is important

Permutations of n distinct objects

Permutations of n objects where one object is repeated s1 times, another is repeated s2 times, and so on

Order is not important

Ways to arrange 10 students at 10 desks: 10! 5 3,628,800

Permutations of n distinct objects taken r at a time

COMBINATIONS

n!

Combinations of r objects taken from a group of n distinct objects

n! (n 2 r )!

P 5}

n r

Ways to arrange 8 students at 10 desks: 10! 2!

} 5 1,814,400

n! s1! p s2! p . . . p sk!

}

Distinguishable permutations of the letters in STUDENTS: 8! 2! p 2!

} 5 10,080

n! (n 2 r )! p r !

C 5}

n r

Ways to choose 8 students from a set of 10 students: 10! 2! p 8!

} 5 45

Chapter Summary

729

10

FLORIDA

CHAPTER CHAPTER CH H REVIEW Go to thinkcentral.com Vocabulary Practice

REVIEW KEY VOCABULARY • sequence, p. 678

• common difference, p. 686

• permutation, p. 713

• terms of a sequence, p. 678

• arithmetic series, p. 688

• factorial, p. 713

• series, p. 680

• geometric sequence, p. 695

• combination, p. 719

• summation notation, p. 680

• common ratio, p. 695

• Pascal’s triangle, p. 721

• sigma notation, p. 680

• geometric series, p. 697

• binomial theorem, p. 722

• arithmetic sequence, p. 686

• partial sum, p. 703

VOCABULARY EXERCISES 1. Copy and complete: The values in the range of a sequence are called the ?

of the sequence. 2. WRITING How can you determine whether a sequence is arithmetic? 3. Copy and complete: A(n) ? is a selection of r objects from a group of n objects

where the order of the objects selected is not important. 4. Copy and complete: In a(n) ? sequence, the ratio of any term to the

previous term is constant.

REVIEW EXAMPLES AND EXERCISES Use the review examples and exercises below to check your understanding of the concepts you have learned in each lesson of Chapter 10.

10.1

Define and Use Sequences and Series

pp. 678–684

MA.912.D.11.2

EXAMPLE Write the series using summation notation: 3 1 6 1 9 1 . . . 1 27.

Notice that the terms of the series are multiples of 3: the first term is 3(1), the second is 3(2), the third is 3(3), and the last term is 3(9). So, the terms of the series can be written as: a i 5 3i where i 5 1, 2, 3, . . . , 9 The lower limit of summation is 1 and the upper limit of summation is 9. 9

c The summation notation for the series is

∑ 3i. i51

EXERCISES EXAMPLES 4, 5, and 6 on pp. 680–681 for Exs. 5–8

730

Write the series using summation notation. Then find the sum. 5. 7 1 14 1 21 1 . . . 1 42

6. 2 1 4 1 8 1 . . . 1 128

7. 1 1 2 1 3 1 . . . 1 17

8. 1 1 4 1 9 1 . . . 1 121

Chapter 10 Sequences, Series, and the Binomial Theorem

Go to thinkcentral.com Chapter Review Practice

10.2

Analyze Arithmetic Sequences and Series

pp. 686–693

MA.912.D.11.1

EXAMPLE Write a rule for the nth term of the sequence 9, 13, 17, 21, 25, . . . .

The sequence is arithmetic with first term a1 5 9 and common difference d 5 4. So, a rule for the nth term is: an 5 a1 1 (n 2 1)d

Write general rule.

5 9 1 (n 2 1)(4)

Substitute 9 for a1 and 4 for d.

5 5 1 4n

Simplify.

EXERCISES EXAMPLES 2, 3, 4, and 5 on pp. 687–689 for Exs. 9–16

Write a rule for the nth term of the arithmetic sequence. 9. 8, 5, 2, 21, 24, . . .

10. d 5 7, a 8 5 54

11. a 4 5 27, a11 5 69

Find the sum of the series. 15

12.

26

∑ (3 1 2i)

13.

i51

∑ (25 2 3i)

30

22

14.

i51

∑ (6i 2 5)

15.

i51

∑ (284 1 8i)

i51

16. COMPUTER Joe buys a $600 computer on layaway by making a $200 down

payment and then paying $25 per month. Write a rule for the total amount of money paid on the computer after n months.

10.3

Analyze Geometric Sequences and Series

pp. 695–701

MA.912.D.11.3

EXAMPLE

7

Find the first term a 1 and the sum of the series

∑ 5(3)

i21

.

i51

The first term of the series is a1 5 5(3) 1 2 1 5 5. Notice that the terms are 5(3) 0, 5(3) 1, 5(3) 2, and so on, so the series is geometric with common ratio r 5 3. 1 2 r7 S7 5 a1 }

1 12r 2

Write rule for S7.

1 2 37 55 }

Substitute 5 for a1 and 3 for r.

5 5465

Simplify.

1 123 2

EXERCISES EXAMPLES 2, 3, 4, and 5 on pp. 696–698 for Exs. 17–23

Write a rule for the nth term of the geometric sequence. Then find a 10. 17. 256, 64, 16, 4, 1, . . .

18. r 5 5, a2 5 200

19. a1 5 144, a 3 5 16

Find the sum of the series. 6

20.

∑ 3(5)i 2 1

i51

9

21.

∑ 8(2)i 2 1

i51

5

22.

∑ 15 1 }23 2i 2 1

i51

7

23.

∑ 40 1 }12 2i 2 1

i51

Chapter Review

731

10

FLORIDA

10.4

CHAPTER CHAPTER CH H REVIEW

Find Sums of Infinite Geometric Series

pp. 703–709

MA.912.D.11.4

EXAMPLE `

Find the sum of the series

∑ 1 }45 2i 2 1, if it exists.

i51

4 For this geometric series, a1 5 1 } 2

121

5

4. 5 1 and the common ratio is r 5 } 5

Because r < 1, the sum of this series exists. a 12r

1 1 5 5. 5} The sum is S 5 }

4 12} 5

EXERCISES EXAMPLES 2, 4, and 5

Find the sum of the infinite geometric series, if it exists.

on pp. 704–705 for Exs. 24–40

24.

`

`

∑ 3 1 }58 2i 2 1

25.

i51

`

26.

i51

`

28.

∑ 21 }34 2 i 2 1

29.

i51

∑ 4(1.3)i 2 1

i51

`

27.

i51

`

∑ 5(0.8)i 2 1

∑ 151 }29 2 i 2 1

i51

`

30.

∑ 20.2(0.5)i 2 1

i51

∑ 71 2}34 2i 2 1 `

31.

∑ 4(20.2)i 2 1

i51

Write the repeating decimal as a fraction in lowest terms. 32. 0.666. . .

33. 0.888. . .

34. 0.4545. . .

35. 0.001001. . .

36. 0.546546546. . .

37. 39.3939. . .

38. 0.3787878. . .

39. 0.7838383. . .

40. BALL BOUNCE You drop a ball from a height of 8 feet. Each time it hits the

ground, it bounces to 40% of its previous height. Find the total distance traveled by the ball.

10.5

Apply the Counting Principle and Permutations

pp. 711–718

Preparation for MA.912.A.4.12

EXAMPLE

An ice skating competition features 8 skaters. How many different ways can the skaters finish the competition? How many different ways can 3 of the skaters finish first, second, and third? As each skater finishes, one fewer remains. There are 8! ways the skaters can finish the competition. 8! 5 8 p 7 p 6 p 5 p 4 p 3 p 2 p 1 5 40,320 The number of ways that 3 of the skaters can finish first, second, and third is the number of permutations of 8 objects taken 3 at a time, 8P 3. 8! 8! 5 336 P 5} 5}

8 3

732

(8 2 3)!

5!

Chapter 10 Sequences, Series, and the Binomial Theorem

Go to thinkcentral.com Chapter Review Practice

EXERCISES EXAMPLES 4 and 5 on pp. 713–714 for Exs. 41–46

41. MOVIES A family of five goes to see a movie. They sit in a single row. How

many ways can the members of the family be seated? 42. PHOTOGRAPHY You are placing 12 pictures on separate pages in an album.

How many different ways can you order the 12 pictures in the album? How many different ways can 4 of the 12 pictures be placed on the first 4 pages? Find the number of permutations. 43.

10.6

P

44.

9 1

P

45.

5 5

P

6 3

P

46.

Use Combinations and the Binomial Theorem

10 2

pp. 719–726

MA.912.A.4.12

EXAMPLE

You must write reports on 3 of the 12 most recent Presidents of the United States. If the order in which you do the reports is not important, how many different reports are possible? The number of possible combinations of reports is: 12! p 11 p 10 p 9! 1320 C 5} 5 12 }} 5 } 5 220

12 3

9! p 3!

9! p 3!

6

EXAMPLE Use the binomial theorem to expand (x 1 5y)4. (x 1 5y)4 5 4C 0x4(5y) 0 1 4C1x 3 (5y)1 1 4C2x2 (5y)2 1 4C3x1(5y) 3 1 4C4x0 (5y)4 5 (1)(x4)(1) 1 (4)(x 3)(5y) 1 (6)(x2)(25y 2) 1 (4)(x)(125y 3) 1 (1)(1)(625y4) 5 x4 1 20x 3y 1 150x2y 2 1 500xy 3 1 625y4

EXERCISES EXAMPLES 1, 3, 5, and 6 on pp. 719–722 for Exs. 47–57

Find the number of combinations. 47.

C

9 2

48.

C

7 1

49.

C

5 3

50.

C

13 5

Use the binomial theorem to write the binomial expansion. 51. (t 1 3) 6

52. (2a 1 b2)4

53. (w 2 8v)4

54. (r 3 2 4s) 5

55. KARATE There are 20 students in a karate class. During a recent lesson, the

instructor asked two students to demonstrate a self-defense move. How many different pairs of students could have been chosen? 56. ICE CREAM An ice cream vendor sells 15 flavors of ice cream. You want to

sample at least 4 of the flavors. How many different combinations of ice cream flavors can you sample? 57. SIGHTSEEING A school group is planning a trip to New York City. Students

have expressed an interest in visiting 10 different locations, but the group can visit at most 4 of these. How many different combinations of locations can the group visit? Chapter Review

733

10

FLORIDA

CHAPTER CHAPTER CH H TEST

Tell whether the sequence is arithmetic, geometric, or neither. Explain. 1. 5, 9, 13, 17, . . .

2. 3, 6, 12, 24, . . .

5, 5, . . . 3. 40, 10, } } 2 8

4. 4, 7, 12, 19, . . .

7. an 5 7n 3

n 8. an 5 } n13

Write the first six terms of the sequence. 5. an 5 3n 2 5

6. an 5 6 2 n 2

Write the next term of the sequence, and then write a rule for the nth term. 9. 5, 11, 17, 23, . . .

10. 3, 15, 75, 375, . . .

6, 7 , 8 , 9 , . . . 11. } } } } 5 10 15 20

12. 1.6, 3.2, 4.8, 6.4, . . .

Find the sum of the series. 48

13.

∑i

28

14.

i51 5

17.

∑ 9(2)i 2 1

10

∑ n2 n51

∑ (4i 2 9)

15.

i51

6

18.

i51

19

16.

i51

`

∑ 12 1 }13 2i 2 1

∑ 8 1 }34 2i 2 1

19.

i51

∑ (2i 1 5) `

20.

i51

3 i21 ∑ 20 1 } 10 2

i51

Write the repeating decimal as a fraction in lowest terms. 21. 0.111. . .

22. 0.464646. . .

23. 0.187187187. . .

24. 0.3252525. . .

Find the number of permutations or combinations. P

26.

8 3

27.

12 7

P

28.

17

C

30. 7C 7

31.

18 4

C

32.

9 5

25.

5 2

29.

4 3

P

P10

C

Use the binomial theorem to write the binomial expansion. 33. (x 1 5) 3

35. (s 1 t 2)4

34. (3a 2 3) 5

36. (c 3 2 2d2)6

37. QUILTS Use the pattern of checkerboard quilts shown.

n 5 1, an 5 1

n 5 2, an 5 2

n 5 3, an 5 5

n 5 4, an 5 8

a. What does n represent for each quilt? What does an represent? b. Make a table that shows n and an for n 5 1, 2, 3, 4, 5, 6, 7, and 8. n 2 1 1 [1 2 (21) n ] to find a for n 5 1, 2, 3, 4, 5, 6, 7, c. Use the rule an 5 } } n 4 2 and 8. Compare these values with the results in your table. What can you

conclude about the sequence defined by this rule? 38. GOVERNMENT There are 15 members on a city council. On a recent agenda

item, 8 of the council members voted in favor of a budget increase for city park improvements and the rest voted against. How many combinations of council members could have voted in favor of the budget increase? How many combinations of council members could have voted against the increase?

734

Chapter 10 Sequences, Series, and the Binomial Theorem

COLLEGE ENTRANCE EXAM PRACTICE FOCUS ON SAT STUDENT-PRODUCED RESPONSES °

Some questions on the SAT require you to enter your answer in a special grid. Your answers must be positive integers, fractions, or decimals. You cannot enter negative numbers or mixed numbers in the grid.

£ Ó Î { x È Ç n ™

HOT TIP

Some questions may have multiple answers; in these cases you may enter any one correct answer. If the solution is an inequality, be sure that you choose a number from the solution region.

You may want to time yourself as you take this practice test. It should take you about 9 minutes to complete.

1. Two terms of a geometric sequence are

4 . What is a ? a3 5 36 and a8 5 } 5 27

2. The first term of a sequence is 2, and the nth term is defined to be 3n 2 1. What is the average of the 7th, 10th, and 12th terms?

° ä £ Ó Î { x È Ç n ™

° ä £ Ó Î { x È Ç n ™

° ä £ Ó Î { x È Ç n ™

5. What is the coefficient of the seventh term in the

expansion of (x2 2 1)12 ?

3

6. What is the sum of the series

n 2 1? ∑} n51n 1 1

7. A manager is scheduling the interviews for

3. The first term of an arithmetic sequence is 25. If the common difference is 4, what is the 7th term of the sequence?

6 people who are applying for a job. The manager has 4 different time slots available for the interviews. How many different interview schedules are possible?

4. Your CD player can hold 6 CDs. You have 10 CDs to choose from, one of which is your favorite and is always in your player. How many ways can the player be filled if order does not matter?

8. What is the sum of the infinite geometric series 3 1 9 2 27 1 …? 22} } } 2 8 32

College Entrance Exam Practice

735

10

FLORIDA

TEST TACKLER Strategies for MULTIPLE CHOICE QUESTIONS Some of the information you need to solve a context-based multiple choice question may appear in a table, a diagram, or a graph.

PROBLEM 1 The frequencies (in hertz) of the notes on a piano form a geometric sequence. The frequencies of G (labeled “8”) and A (labeled “10”) are shown in the diagram. What is the approximate frequency of E flat (labeled “4”)? A 247 Hz

B 311 Hz

C 330 Hz

D 554 Hz

2 1

7

4 3

5

6

9 8

392 Hz

11 10

12

440 Hz

Plan INTERPRET THE DIAGRAM The diagram gives you the frequencies of the 8th and 10th notes. Use these frequencies to find the frequency of the 4th note.

STEP 1 Write a system of equations.

STEP 2 Solve the system of equations to find the values of r and a1.

Solution Let an be the frequency (in hertz) of the nth note. Because the frequencies form a geometric sequence, a rule for an has the form an 5 a1r n 2 1. From the diagram, a 8 5 392 and a10 5 440. Use these values to write a system of equations. a 8 5 a1r 8 2 1

392 5 a1r 7

Equation 1

a10 5 a1r 10 2 1

440 5 a1r 9

Equation 2

392 a1 5 } 7

Solve Equation 1 for a1.

r

392 p r 9 440 5 } 7

Substitute } for a1 in Equation 2. 7

440 5 392r 2

Simplify.

r

1.12 ø r

2

392 r

Divide each side by 392.

1.06 ø r

Take positive square root of each side.

Find a1 by substituting the value of r into revised Equation 1. 392 5 392 ø 261 a1 5 } } 7 7 r

STEP 3 Write a rule for the nth term and find a4.

(1.06)

A rule for the sequence is an 5 a1r n 2 1 5 261(1.06) n 2 1. So, a 4 5 261(1.06) 3 ø 311. c The correct answer is B. A B C D

736

Chapter 10 Sequences, Series, and the Binomial Theorem

PROBLEM 2 The first 4 terms of an infinite arithmetic sequence are shown in the graph. Which rule describes the nth term in the sequence? A an 5 2n 2 5

B an 5 2n 1 5

C an 5 5n 2 2

D an 5 n 1 5

an (4, 3) 1

(3, 1) n

1

(2, 21) (1, 23)

Plan INTERPRET THE GRAPH In order to find a rule for the sequence, you must first use the graph to write the terms of the sequence.

STEP 1 Write the terms of the sequence.

Solution The points shown in the graph are: (1, 23), (2, 21), (3, 1), (4, 3) Therefore, the sequence is 23, 21, 1, 3, . . . .

STEP 2 Find the first term and the common difference.

STEP 3 Write a rule for the nth term.

The first term a1 of the sequence is 23. Because each term after the first is 2 more than the previous term, the common difference d is 2. an 5 a1 1 (n 2 1)d

Write general rule for an arithmetic sequence.

5 23 1 (n 2 1)2

Substitute 23 for a1 and 2 for d.

5 23 1 2n 2 2

Distributive property

5 2n 2 5

Simplify.

c The correct answer is A. A B C D

PRACTICE In Exercises 1 and 2, use the graph in Problem 2. 1. What is the value of a15 ?

A 235

B 25

C 30

D 165

2. Which statement is true about the sequence that is graphed?

F The sum of the first 14 terms is 140. G The value of a20 is 40. H The common difference is d 5 5. I The ratio of any term to the previous term is constant.

Test Tackler

737

10

MASTERING the STANDARDS

FLORIDA

MULTIPLE CHOICE 1. The diagram shows the bounce heights of

a basketball and a baseball dropped from a height of 10 feet. On each bounce, the basketball bounces to 36% of its previous height, and the baseball bounces to 30% of its previous height. About how much greater is the total distance traveled by the basketball than the total distance traveled by the baseball? 10 ft

In Exercises 4 and 5, use the information below. Cheryl is researching her lineage for a history project. So far, she has created a family tree for three generations, as shown below. Cheryl is only including relatives from whom she is directly descended. Siblings are not included. Maternal grandmother

Maternal grandfather

Paternal grandmother

Paternal grandfather

10 ft Father

Mother

Cheryl

3.6 ft 1 3.6 ft

4. Assume that Cheryl is in generation 1, her

3 ft 1 3 ft

1.3 ft 1 1.3 ft

Basketball

0.9 ft 1 0.9 ft

Baseball

A 1.34 feet

B 2.00 feet

C 2.62 feet

D 5.63 feet

In Exercises 2 and 3, use the information below. The table shows the numbers of Democratic and Republican U.S. Presidents in office from 1853 to 2009 by the region of their birth. Midwest

Northeast

South

West

D

3

6

4

0

R

9

5

3

1

You are to report on 4 of these Presidents for a history project. 2. Which expression describes the combinations

of 4 Presidents all from the same party? F H

C p 18C 4

13 4

C p 31C 4

2 1

G I

C 1 18C 4

13 4

C 1 31C4

2 1

3. Which expression describes the combinations

of 2 Democrats who were born in the South and 2 Republicans born in the Midwest? A C

738

C p 9C 2

4 2

C p 4C2

13 4

B D

C 1 9C 2

4 2

C 1 4C2

13 4

Chapter 10 Sequences and Series

parents are in generation 2, and so on. Let an be the number of relatives in generation n. What is a rule for an ? F an 5 n 1 2

G an 5 2 n 2 1

H an 5 2 n

I an 5 2 n 1 1

5. Cheryl creates a family tree with 8 generations

of her family. How many people are in her family tree? A 8

B 64

C 128

D 255

In Exercises 6 and 7, use the diagram of a stack of blocks.

6. Which rule describes the number of blocks in

the nth layer, where n 5 1 represents the top layer? F an 5 n 1 1 H an 5 2n

2

G an 5 n(n 1 1) I an 5 n 2 1 1

7. Which sum gives the number of blocks shown? 20

A

∑ (i 1 1)

4

B

i52

i51

20

C

∑ i(i 1 1)

i52

∑ (i 1 1) 4

D

∑ i(i 1 1)

i51

Florida Test Practice Go to thinkcentral.com

GRIDDED RESPONSE

SHORT RESPONSE

8. An online account requires that you choose

a password that has from 6 to 8 characters. The characters can be letters or digits. Two examples are shown below. 4 T F I V E

L O G B A S E 2

How many times as many passwords are there if repetition is allowed than if repetition is not allowed? Round your answer to one decimal place. 9. Two terms of a geometric sequence are

a3 5 12 and a5 5 48. What is the value of a1? 10. Write the repeating decimal 0.151515. . . as a

fraction in lowest terms. 11. Find the number of distinguishable

permutations of the letters in the word WEEKEND. 12. What is the coefficient of x 2 in the expansion of

(4x 2 1) 9?

13. You receive an e-mail that you are to forward

to 10 of your friends. There were 10 recipients in the first round, 100 recipients in the second round, and so on. By the time you receive the e-mail, it has already been sent to just over 100 million people. What round of recipients must you be in? Explain your reasoning. 14. The number of diagonals in a convex polygon 1 n(n 2 3) where is given by the formula dn 5 } 2 n is the number of sides of the polygon (n ≥ 3).

Write the first six terms of the sequence given by the formula. Then tell whether the sequence is arithmetic, geometric, or neither. Explain. 15. For which are there more possible choices:

8 students from 24 or 16 students from 24? Explain without performing calculations. 16. During a baseball season, a company pledges

a donation to a charity of $5000 plus $100 for every home run hit by the local team. Does it make more sense to represent this situation using a sequence or a series? Explain.

EXTENDED RESPONSE 17. A running track is shaped like a rectangle with two

semicircular ends, as shown. The track has 8 lanes that are each 1.22 meters wide. The lanes are numbered from 1 to 8 starting from the inside lane. The length of each red line segment that extends from the center of the left semicircle to the inside of a lane is called the lane’s curve radius.

1.22 m

83.4 m 36.5 m

a. Is the sequence formed by the curve radii arithmetic,

geometric, or neither? Explain. b. Write a formula for the sequence from part (a).

Not drawn to scale

c. World records must be set on tracks for which the curve radius in the

outside lane is at most 50 meters. Does the track shown meet this requirement? Explain. 18. A 9 digit Social Security Number (SSN) is assigned to every citizen of the

United States. The digits are 0 to 9 and can repeat. a. How many possible SSNs are there? b. In 2008, about 442 million SSNs had been assigned. If 6 million new

SSNs are assigned each year, about what year will the Social Security Administration run out of new numbers? c. After the Social Security Administration runs out of new numbers to

assign, what do you think it should do? Explain.

Mastering the Standards

739

10

CUMULATIVE REVIEW

Chapters

1–10

Solve the equation. Check your solution(s). 1. 5x 1 24 5 11 2 2x (1.3)

2. 4x 2 7 5 13 (1.7)

3. x2 2 12x 1 35 5 0 (4.3)

4. 2x2 2 5x 1 5 5 0 (4.8)

5. x 3 1 3x 2 2 18x 5 40 (5.6)

6. Ï x 2 2 5 x 2 4 (6.6)

7. 4x 2 5 5 3 (7.6)

x 1 3 5 x (8.6) 8. } } x12 3x 1 1

x 2 4 1 2 5 2x 2 3 (8.6) 9. } } x23 x23

}

Graph the equation. 10. 3x 2 y 5 5 (2.3)

1 x 1 3y 5 24 (2.3) 11. } 2

12. y 5 x 1 3 2 8 (2.7)

13. y 5 x2 2 6x 2 27 (4.1)

14. y 5 22(x 1 6)(x 2 1) (4.2)

15. y 5 (x 2 3)2 1 4 (4.2)

16. y 5 (x 1 1)2 (x 2 2) (5.8)

17. y 5 Ï x 1 6 (6.5)

19. y 5 3 p 4x 2 2 (7.1)

1 20. y 5 12 } 8

22. y 5 ln (x 2 2) (7.4)

2 1 5 (8.2) 23. y 5 } x23

6 24. y 5 } (8.3) x2 2 4

26. 6x 2 1 7x 2 20 (4.4)

27. x 3 1 8x 2 2 4x 2 32 (5.4)

3}

}

x

1 2

18. y 5 Ïx 2 2 (6.5) 21. y 5 4e x (7.3)

(7.2)

Factor the expression. 25. 2x2 2 20x 2 48 (4.4)

Find the inverse of the function. (6.4) 28. f(x) 5 6x 2 1

29. f(x) 5 x 3 2 5

30. f(x) 5 x5

32. log3 4 1 2 log3 7

33. 5 log x 1 log y 2 3 log z

Condense the expression. (7.5) 31. 3 ln x 2 ln 5

The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find the value of y when x 5 28. (8.1) 34. x 5 3, y 5 6

35. x 5 24, y 5 9

1 36. x 5 4, y 5 } 8

2 37. x 5 9, y 5 } 5

Classify the conic section and write its equation in standard form. Then graph the equation. (9.6) 38. x 2 1 y 2 1 12x 2 4y 1 15 5 0

39. 4x 2 2 16y 2 2 56x 1 160y 2 268 5 0

40. y 2 1 6x 1 4y 1 16 5 0

41. 2x2 1 3y 2 1 4x 1 12y 2 14 5 0

Graph the equation. 2 x 2 1 y 5 1 (9.4) 42. } } 36 4

y2 x 2 5 1 (9.5) 43. } 2 } 49 100

44. (x 2 3)2 5 16y (9.6)

Find the sum of the series. 6

45.

∑ 3i2 (10.1)

16

46.

i51

∑ (22 1 i) (10.2)

i51

12

47.

∑ 1 }2 2 i51 3

`

i21

(10.3)

48.

∑ 5 1 }13 2

i21

i51

Find the number of permutations or combinations. 49.

740

P (10.5)

9 3

Cumulative Review: Chapters 1–10

50.

P (10.5)

16 5

51. 7C2 (10.6)

52.

C (10.6)

6 6

(10.4)

53. FUNDRAISER You are organizing a school fundraiser that involves selling

holiday cookies and decorative calendars. You want to raise $2400. You charge $2 for a bag of cookies and $7 for a calendar. Write and graph an equation to represent the situation. If you sell 200 calendars, how many bags of cookies do you need to sell in order to meet your goal? (2.4) 54. BASKETBALL The price of admission to a high school basketball game is

$5 for adults and $2 for students. At a game for which 650 tickets were sold, the total income from ticket sales was $2500. Write and solve a linear system to find the numbers of adults and students who attended the game. (3.2) 55. TENNIS While serving, a tennis player strikes the ball at a height of 9 feet

above the court. The initial downward velocity of the ball is 16 feet per second. How long does it take the ball to strike the court on the opponent’s side? (Hint: Use the function h 5 216t 2 1 v 0t 1 h0.) (4.8) 56.

GEOMETRY A designer is creating a kit for making sand castles. The designer wants one of the molds to be a cone that will hold 75π cubic inches of sand. What should the dimensions of the cone be if the height should be 4 inches more than the radius of the base? (5.6)

r14

r

57. DISCOUNTS A store is having a sale in which you can take $50 off the cost

of any television in the store. The store also offers 15% off your purchase if you open a charge account. Use composition of functions to write a new function that gives the sale price of a television that originally costs t dollars if $50 is subtracted before the 15% discount is applied. Then find the sale price of a television that originally cost $480. (6.3) 58. ELECTRICITY The current I (in amperes) required for an electrical } P where P is the power (in watts) and R is the appliance is given by I 5 } R

Î

resistance (in ohms). Find the power consumed by a portable hair dryer for which I 5 17 amperes and R 5 6.5 ohms. (6.6) 59. ACCOUNT BALANCE You deposit $4500 in a savings account that pays

2.75% annual interest compounded monthly. Find the account balance after 5 years. (7.1) 60. DEPRECIATION Rachel buys a new car for $18,600. The value of the car

decreases by 15.5% each year. Estimate when the car will have a value of $8000. (7.2) 61.

GEOMETRY Steve is a lifeguard at a pond. The pond is approximately circular with a diameter of 330 feet. He ropes off a section of the pond for swimming. The rope forms a chord of the circle with a maximum distance of 50 feet from the edge of the pond. What is the length of the rope? (9.3)

62. SALARY An accountant takes a job that pays an annual salary of $31,000

for the first year. The employer offers a $1600 raise for each of the next 8 years. Write a rule for the accountant’s salary in the nth year. What will the accountant’s salary be in the 9th year? (10.2) Cumulative Review: Chapters 1–10

741