April 30th copyright2009merrydavidson

Happy Birthday to:

4/25 Lauren Cooper

9.1 Sequences & Series

SEQUENCE:A list that is

ordered so that it has a 1st term, a 2nd term, a 3rd term and so on. example: 1, 5, 9, 13, 17, …

a1 = 1; a2 = 5; a3 = 9, etc.

The nth term is denoted by: an

The domain of a sequence is the set of positive integers.

The nth term is used to GENERALIZE about other terms.

The three dots mean that this sequence is INFINITE.

example: 1, 5, 9, 13, 17, …

example: 2, -9, 28, -65, 126

This is a FINITE sequence.

“Series” uses + signs.

Arithmetic Sequence Arithmetic Series

 

3, 8, 13, 18, 23 3 + 8 + 13 + 18 + 23

Given a “rule” for a sequence,

find the 1st 5 terms.

2

4

14 1 7

2f

211

2nf x

2

1

1 11 1

2 2f 2

2

12 1 1

2f

2

3

1 73 1

2 2f

2

5

1 235 1

2 2f

1 7 23,1, ,7,

2 2 2

EXAMPLE 1:

Example 2:

1

2

n

nf

1 1 1 1, , ,

2 4 8 16

Write the first 4 terms of the sequence.

Example 3. Write the first six terms of the sequence if

1 k+11, a 2ka a

1,3,5,7,9,11

Factorial Notation

n! = n(n – 1)(n – 2)…1

Special case: 0! = 1

8! 8 7 6 5 4 3 2 1 8 math/prb/4/enter

= 40,320

Factorial Notation

504

n! = n(n – 1)(n – 2)…1

Special case: 0! = 1

9!

5!3!9 8 7 6 5!

5!3!

Summation Notation

1

n

k

rule

The Greek letter sigma, instructs you to add up the terms of the sequence.

52

1

( 1)i

i

Example of sigma notation

Example 4.

52

1

( 1)i

i

Starting with an i value of 1 and ending with an i value of 5, write the series, then add.

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

2 + 5 + 10 + 17 + 26 = 60

Find the sum of:

Example 5.

Starting with an k value of 3 and ending with a k value of 6, write the expanded sum.

62

3

( 1)k

k

2 2 2 2(3 1) (4 1) (5 1) (6 1) Notice: k=3 to k =6 is 4 terms

10 + 17 + 26 + 37 = 90

Find the sum of:

Example 6.

-1 + 0 + 1 + 8 + 27 = 35

43

0

( 1)j

j

Find the sum of:

Notice there are 5 terms here because you are starting at zero.

• A sequence uses comma’s

• A series uses + signs

• Summation notation uses sigma sign

1, 5, 9, 13, 17, …

The common difference is 4

When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. In other words, the terms increase (or decrease) by adding a fixed quantity “d”.

Is this sequence arithmetic?

Example 7:

2, -4, 8, -16, 32…

No because we are multiplying by -2 each time.

Is this sequence arithmetic?

Example 8:

-5, 7, 19, 31,…

yes because we are adding 12 each time.

Is the sequence defined by

Sn= 3n + 5 arithmetic?

Example 9:

Let n = 1, n = 2, n = 3, etc to generate the sequence.

8, 11, 14, 17…

yes because we

are adding 3 each time.

Notice that the common

difference is the “slope” of the function.

Therefore linear functions are arithmetic!

Is the sequence defined by

Sn= 4 - n arithmetic?

Example 10:

Let n = 1, n = 2, n = 3, etc to generate the sequence.

3, 2, 1, 0, …

yes because we

are subtracting 1 each time.

d = -1

a1 = 3Therefore linear functions are arithmetic!

Formula for the nth term of Arithmetic Sequence:

“a” is the first term and “d” is the common difference

1 ( 1)na a n d

11) Write the nth term of the sequence 2, 7, 12, 17,…..

Step 1: find the common difference 5

Step 2: write down the formula

1 ( 1)na a n d Step 3: fill in the formula with what you know

2 ( 1)5na n

2 5 5na n

5 3na n

12) Write the nth term of the sequence -12, -9, -6, …..

Step 1: find the common difference 3

Step 2: write down the formula

1 ( 1)na a n d Step 3: fill in the formula with what you know

12 ( 1)3na n

12 3 3na n

3 15na n

Use when you know the first term, number of terms and common difference

Use when you know first term, last term, and number of terms

1

1[2 ( 1) ]

2nS n a n d 1

1( )

2n nS n a a

Notice: Both formulas need first term and number of terms.

SUMMATION FORMULAS:

13) Find the sum of the first 12 terms of:

• Find “d”.

.

4 8 12 16 20

d = 4• Pick which formula

you want to use.

1

1[2 ( 1) ]

2nS n a n d

• Plug and Chug. 12

112[2 4 (12 1) 4 ]

2S

12 6[8 (11) 4 ]S

12 312S

14) Find the indicated partial sum of:• Find the 1st term.

.

a1 = -1

• Pick which formula you want to use. • Plug and Chug.

25

0

(2 1)j

j

• Find the 2nd term.

a2 = 1• Find “d”. d = 2

• Find the last term.

a26 = 51

1

1( )

2n nS n a a

1(26)( 1 51)

2nS

650nS