geometric sequences and series

Geometric Sequences and Series

Section 9-3

2

Objectives

• Recognize, write, and find nth terms of geometric sequences

• Find the nth partial sums of geometric sequences

• Find the sum of an infinite geometric sequence

3

Definition of a Geometric Sequence

• A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

4

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

a1, a2, a3, a4, . . . , an, . . .

The first three terms of the sequence an = 2n2 are

a1 = 2(1)2 = 2

a2 = 2(2)2 = 8

a3 = 2(3)2 = 18.

finite sequence

terms

5

A sequence is geometric if the ratios of consecutive terms are the same.

2, 8, 32, 128, 512, . . .

geometric sequence

The common ratio, r, is 4.

82

4

328

4

12832

4

512128

4

6

General Term of a Geometric Sequence

• The nth term (the general term) of a geometric sequence with the first term a1 and common ratio r is

• an = a1 r n-1

7

The nth term of a geometric sequence has the form

an = a1rn - 1

where r is the common ratio of consecutive terms of the sequence.

15, 75, 375, 1875, . . . a1 = 15

The nth term is: an = 15(5)n-1.

75 515

r

a2 = 15(5)

a3 = 15(52)

a4 = 15(53)

8

Example: Find the 9th term of the geometric sequence

7, 21, 63, . . .

a1 = 7

The 9th term is 45,927.

21 37

r

an = a1rn – 1 = 7(3)n – 1

a9 = 7(3)9 – 1 = 7(3)8

= 7(6561) = 45,927

9

The Sum of the First n Terms of a Geometric Sequence

r

raS

n

n

1

)1(1

The sum, Sn, of the first n terms of a geometric sequence is given by

in which a1 is the first term and r is the common ratio.

10

Example

5314404

)5314411(4

)3(1

))3(1(4

1

)1(

1

)1(

12121

12

1

r

raS

r

raS

n

n

• Find the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108, ...Solution:

11

The sum of the first n terms of a sequence is represented by summation notation.

1 2 3 41

n

i ni

a a a a a a

index of summation

upper limit of summation

lower limit of summation

5

1

4n

n

1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364

12

The sum of a finite geometric sequence is given by

11 1

1

1 .1

n nin

i

rS a r ar

5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?

n = 8

a1 = 5

1

81 11

221

5n

nrS ar

5210r

1 25651 2 2555

1 1275

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The Sum of an Infinite Geometric Series

If -1<r<1, then the sum of the infinite geometric series

a1+a1r+a1r2+a1r3+…

in which a1 is the first term and r s the common ration is given by

r

aS

11

If |r|>1, the infinite series does not have a sum.

14

Example: Find the sum of

1

1a

Sr

1 13 13 9

13

r

3

1 13

3 31 413 3

The sum of the series is 9 .4

3 934 4

15

...16

1

8

1

4

1

2

1

r

aS

11

21

21

1S

121

21

S

16

...64

3

32

3

16

3

8

3

r

aS

11

21

83

1 S

4

1

23

83

S

17

Homework

• WS 13-5