Mrs. Turner’s Precalculus – page 0

Sequences and Series Binomial Theorem

This is the Golden Spiral – a special ratio that occurs in nature

Name: ________________________________ Period: _____________

Mrs. Turner’s Precalculus – page 1

8.1 – Sequences and Series Notes  What is a sequence? A sequence is a collection of terms: a1 , a2 , a3 ,....... an , ...... Write the first four terms of the sequences given by 1. an

3n 2

What is a30 ?

2. a n

3 ( 1) n

What is a100 ?

3. an

( 1) n 2n 1

What is a15 ?

 Recursive sequences – one of more of the first few terms will be given. The other terms in the sequence are found using the previous terms 4. Write the first six terms of the Fibonacci sequence that is defined as follows: a0 1 , a1 1, ak ak 2 ak 1

5. Write the first five terms of the sequence that is defined recursively.

 Factorial Notation: n! 1 2 3 .....(n 1) n

Mrs. Turner’s Precalculus – page 2 By definition, 0! = 1

Find the following values: 1! = 2! = 3! = 4! = 5! = 6. Write the first five terms of the sequence given by an

Evaluate each factorial expression 8! 2! 6! 7. 8. 2! 6! 3! 5!

2n (Begin with n = 0) n!

9.

n! ( n 1)!

 Summation Notation (Sigma Notation) is a convenient way to represent the sum of the terms of a sequence

n

ai

a1 a 2

a3 .... a n

The i is the starting index (lower limit)

i 1

The n is the ending index (upper limit) Find the sum of the following: 10.

5

3i i 1

Mrs. Turner’s Precalculus – page 3 6

11.

(1 k 2 ) k 3

8

12. n

1 0 n!

 Series – the sum of the terms of a sequence a) The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by

n

a i = a1 a2

a3 .... an

i 1

b) The sum of all the terms of the infinite sequence is called an infinite series and is denoted by

a i = a1 a2

a3 .... ai .......

i 1

13. For the series i

3 , find the third partial sum and the sum i 1 10

Mrs. Turner’s Precalculus – page 4

8.2 – Arithmetic Sequences and Partial Sums  A sequence is arithmetic if the terms have a common difference that we will call d Are the following sequences arithmetic? 1. 7, 11, 15, 19, ….. , 4n + 3 2. 1, 4, 9, 16, ……

4. Find the first 4 terms of the arithmetic sequence a1

3. 2, -3, -8, -13, …. , 7 – 5n

7, d

 The nth term of an arithmetic sequence has the form an difference and c a1 d

7

dn c where d is the common

5. Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2

6. The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first several terms of this sequence.

7. Find the seventh term of the arithmetic sequence whose first two terms are 2 and 9.

8. Write an equation for the nth term for the arithmetic sequence -5, -3, -1, 1,…….

Mrs. Turner’s Precalculus – page 5 9. Write an equation for the nth term for the arithmetic sequence -5, 3, 11, 19,……..

 The sum of a Finite Arithmetic Sequence: S n

n (a1 an ) 2

Find the sum using S n 10. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

11. Sum of the integers from 1 to 100

12. Find S n if a1 16 , an

13. Find S n if a1

98, n 13

14. Find the first three terms of the series: a1

14 , an

85, S n

121, an

1207

 Finding a partial sum of an arithmetic sequence 15. Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, ……

16. Find the 10th partial sum of 8, 20, 32, 44, …..

5, d

3

Mrs. Turner’s Precalculus – page 6

8.3 – Geometric Sequences and Series Notes  A geometric sequence has a common ratio we will call r 1. Find the next two terms of the geometric sequence -15, -30, -60,……

2. Find the first five terms of the geometric sequence a1

3. Find the 6th term of the geometric sequence a1

5, r

1, r

3

3

4. Find the 12th term of the geometric sequence 96, 48, 24

 The nth term of a geometric sequence has the form an

a1r n 1

5. Write an equation for the nth term of the geometric sequence 1, 4, 16,……

6. Write an equation for the nth term of the geometric sequence 7, -14, 28, …..

7. Find the 15th term of the geometric sequence whose 1st term is 20 and common ratio is 1.05

8. Write the first five terms of the geometric sequence whose 1st term is 3 common ratio is 2

Mrs. Turner’s Precalculus – page 7  The Sum of a Finite Geometric Sequence with a common ration not equal to 1 is given by the 1 rn definition S n a1 1 r Find the sum S n for each geometric series described: 9. a1

11. a1

2, a6

64 , r

6, r

10. a1

2

3, n

7

12. a1

81, an

5, r

3, n

16, r

2 3

9

13. Find the sum of 162 + 54 + 18 + … to 6 terms

14. Find the sum of

8

( 3) n 1 n 1

15. Find the 1st term for the geometric series that has S n

1023 , an

 The sum of an infinite Geometric Series is defined as Sn

768 , r

4

a1 1 r

Find the sum of each infinite geometric series (if it exists) 16. a1

35, r

2 7

17. 2 + 6 + 18 + …..

1 18. 3 4 n 1

n 1

Mrs. Turner’s Precalculus – page 8

Applications of Sequences and Series Notes Example 1: An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

Example 2: A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annuals sales by $7500 each year for 19 years. Assuming that this goal is met, find the total sales during the first 20 years this business is in operation.

Example 3: You go to work for a company that pays $0.01 the first day, $0.02 the second day, $0.04 the third day and so on. If the daily wage keeps doubling, what will your total income be after working (a) 29 days? (b) 30 days? (c) 31 days?

Mrs. Turner’s Precalculus – page 9 Example 4: The revenue (in billions of dollars) for UPS from 1997 to 2002 can be approximated by the model where n is the year, with n =7 corresponding to 1997. Find the terms of this finite sequence and construct a bar graph that represents this sequence.

Example 5: In his first trip bailing hay around a field, a farmer makes 123 bales. In his second trip he makes 11 fewer bales. Because each trip is shorter than the preceding trip, the farmer estimates that the same pattern will continue. Estimate the total number of bales made if there are another six trips around the field.

Example 6: A company buys a fleet of 6 vans for $120,000. During the next 5 years, the fleet will depreciate at a rate of 30% per year. (That is, at the end of each year, the depreciated value is 70% of the value at the beginning of the year). a) Find a formula for the nth term of a geometric sequence that gives the value of the fleet t full years after it was purchased. b) Find the depreciated value of the fleet at the end of 5 full years.

Mrs. Turner’s Precalculus – page 10

8.5 – Binomial Theorem Notes  You have already discovered how to construct Pascal’s Triangle using binomial expansion. In this activity, you raised (x + y) to different powers. Let’s called each of those powers n.  There are several observations you can make about each expansion you did: 1. In each expansion there are ______________ terms 2. In each expansion, x and y have symmetric roles. The powers of x ______________________, while the powers of y __________________________ 3. The sum of the powers of each term is _________. 4. The coefficients _________________ and then ____________________ in a symmetric pattern.  The coefficients in Pascal’s triangle are called ________________________________________.  To find the binomial coefficients, you can use the BINOMIAL THEOREM. Write the expansion for the given expression. 1.

2.

3.

4.

5.

Mrs. Turner’s Precalculus – page 11 Write the expansion for each expression. Simplify your answers. 6.

7.

8.

9.

10.

11.

12.

Mrs. Turner’s Precalculus – page 12

Binomial Theorem – Finding a Specific Term  Using Binomial Theorem to find the Coefficients The coefficient of x n r y r is given by n Cr

n r

n! r is term you want minus 1. (n r )!r !

Remember you can use your calculator. 1. Expand ( x 6)5 using the nCr on the calculator. 5

C0 ( x)5 (6)0

________________+________________+____________________+____________________+_________________+_________________

2. Find the fourth term of ( x 8)10

3. Find the 6th term of (a 2b)8

4. Find the ninth term of (10 x 3 y )12

Mrs. Turner’s Precalculus – page 13 Find the coefficient (a) of the term in the expansion of the binomial. Binomial 12 5. x 3

6. x 2 y

10

7. 3x 2 y

8. x 2

y

9

10

Term ax4

ax 8 y 2

ax 6 y 3

ax 8 y 6