8.1 Sequences and Series

Essential Questions: How do we use sequence notation to write the terms of a sequence? How do we use factorial notation? How do we use summation

notation to write sums?

Definition of a Sequence

An infinite sequence is a function whose domain is the set of positive integers. The function values

are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

1 2 3 4, , , ,...., ,...na a a a a

Finding Terms of a Sequence

□ The first four terms of the sequence given by are

3 2na n

1 13 2 1a

2 23 2 4a

3 33 2 7a

4 3 104 2a

The first four terms of the sequence given by

are

3 ( 1)nna

11 3 ( 1) 3 1 2a

22 3 ( 1) 3 1 4a

33 3 ( 1) 3 1 2a

44 3 ( 1) 3 1 4a

Finding Terms of a Sequence

□ Write out the first five terms of the sequence given by

( 1)

2 1

n

na n

Solution:

1

1 1

( 1) 11

2 1 2 1a

2

2 2

( 1) 1 1

2 1 4 1 3a

3

3

( 1) 1 1

2 1 63 1 5a

4

4 4

( 1) 1 1

2 1 8 1 7a

5

5

( 1) 1 1

2 1 105 1 9a

Finding the nth term of a Sequence

□ Write an expression for the apparent nth term (an) of each sequence.

□ a. 1, 3, 5, 7, … b. 2, 5, 10, 17, …

Solution:

a. n: 1 2 3 4 . . . n

terms: 1 3 5 7 . . . an

Apparent pattern: Each term is 1 less than twice n, which implies that 2 1na n

b. n: 1 2 3 4 … n

terms: 2 5 10 17 … an

Apparent pattern: Each term is 1 more than the square of n, which implies that 2 1na n

Additional Example

□ Write an expression for the apparent nth term of the sequence:

2 3 4 5, , , ,...

1 2 3 4

Solution:

: 1 2 3 4 ...

2 3 4 5 terms: ...

1 2 3 4 n

n n

a

Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that 1

n

na

n

Factorial Notation

□ If n is a positive integer, n factorial is defined by

As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several

nonnegative integers. Notice that 0! is 1 by definition.

! 1 2 3 4... ( 1)n n n

0! 1

1! 12! 1 2 2

3! 1 2 3 6 4! 1 2 3 4 24 5! 1 2 3 4 5 120

The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3,628,800.

Finding the Terms of a Sequence Involving Factorials

□ List the first five terms of the sequence given by

Begin with n = 0.

2

!

n

na n

0

0

2 11

0! 1a

1

1

2 22

1! 1a

2

2

2 42

2! 2a

3

3

2 8 4

3! 6 3a

4

4

2 16 2

4! 24 3a

Evaluating Factorial Expressions□ Evaluate each factorial expression. Make sure you use

parentheses when necessary.a. b. c.

8!

2! 6!2! 6!

3! 5!

!

( 1)!

n

n

Solution:

a.

b.

c.

8! 1 2 3 4 5 6 7 8 7 828

2! 6! 1 2 1 2 3 4 5 6 2

2! 6! 1 2 1 2 3 4 5 6 62

3! 5! 1 2 3 1 2 3 4 5 3

! 1 2 3...( 1)

( 1)! 1 2 3...( 1)

n n nn

n n

Have you ever seen this sequence before?

□1, 1, 2, 3, 5, 8 …□Can you find the next three terms in

the sequence?□Hint: 13, □21, 34□Can you explain this pattern?

The Fibonacci Sequence

□ Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. A well-known example is the Fibonacci Sequence.

□ The Fibonacci Sequence is defined as follows:

0 1 2 11, 1, , where 2k k ka a a a a k

Write the first six terms of the Fibonacci Sequence:

1 1a 0 1a

2 1 02 2 1 1 1 2a a a a

2 1 13 3 2 1 2 3a a a a

2 1 24 4 3 2 3 5a a a a

2 1 35 5 4 3 5 8a a a a

Summation Notation

□Definition of Summation NotationThe sum of the first n terms of a sequence is

represented by

Where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.

1 2 3 41

...n

n ni

a a a a a a

Summation Notation for Sums

□ Find each sum.a. b. c. 5

1

3i

i

62

3

(1 )k

k

8

0

1

!i i

Solution:

a. 5

1

3 3(1) 3(2) 3(3) 3(4) 3(5)

3(1 2 3 4 5) or 3 6 9 12 15

45

i

i

Solutions continued

b.

c.

62 2 2 2 2

3

(1 ) (1 3 ) (1 4 ) (1 5 ) (1 6 )

10 17 26 37

90

k

k

8

0

1 1 1 1 1 1 1 1 1 1

! 0! 1! 2! 3! 4! 5! 6! 7! 8!

1 1 1 1 1 1 1 1 1

2 6 24 120 720 5040 40320 2.71828

i i

Notice that this summation is very close to the irrational number

. It can be shown that as more terms of the sequence whose nth term is 1/n! are added, the sum becomes closer and closer to e.

2.718281828e

How to Input Sums in your

calculator□ Good news! This can all be done using the TI-84 Plus

graphing calculator.□ To enter in example a, hit the following keys:

The following screen will appear: Now hit 5. Then, hit

□ Good news! This can all be done using the TI-84 Plus graphing calculator.

□ To enter in example a, hit the following keys:

The following screen will appear: Now hit 5. Then, hit

The following screen should appear:

Choose 5.

Be sure you are in sequence mode on the calculator!

This is what your calculator screen should look like:

Now type in the sum, the variable, the lower limit, the upper limit, and the increment (default is 1).

Assignment

□Page 587-5882-32 even, 57-59 odd, 62-64 even, 66-

70 even, 80-94 even