Section 8.3

Geometric

Sequences

&

Series

Geometric Sequences & SeriesA sequence in which a set number is multiplied to eachprevious term is called a geometric sequence.

Definition of Geometric Sequence

A sequence is geometric if the ratios of consecutiveterms are the same. So, the sequence a1, a2, a3, a4, …, an, … is geometric if there is a numberr such that

The number r is the common ratio of the sequence.

32 4

1 2 3

. . . , 0aa a

r ra a a

Geometric Sequences & Series

1 2, 4, 8, 16, 32,...

2 1, 5, 9, 13, 17,...

3 1, 1, 1, 1, 1,...

2 2 2 24 2, , , , ,...

3 9 27 81

8, 2

4yes r

5 9,

1 5no because

Examples:

Determine if each of the following is a geometric sequence. If so, determine the common ratio.

1, 1

1yes because r

22 27 181,

2 81 2 327

yes because r

Geometric Sequences & Series

How does one find a certain term of a geometric sequence?

The nth term of a Geometric Sequence

The nth term of a geometric sequence has the form

an = a1rn–1

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

Geometric Sequences & Series

1

2

3

4

5

2

2 3 6

6 3 18

18 3 54

54 3 162

a

a

a

a

a

Examples:

1. What are the first five terms of the geometricsequence whose first termis a1 = 2 and whose commonratio is r = 3.

2. Find the 10th term of the geometric sequence whosefirst term is 8 and whosecommon ratio is ¼ .

10 1

10

9

10

10

10

184

184

18262144

1

32768

a

a

a

a

Geometric Sequences & Series

1

1

1

1

364 3

12

4 3

n

n

n

n

a r

a a r

a

1 3

4 6

1

1

4 1

3

3

3 33

1

251

3125

1 1

3125 251 25

3125 11

125

1

125

n

n

Let a a

Let a a

a a r

r

r

r

r

Examples:

3. Find a formula for thenth term of the followingsequence:4, 12, 36, 108, 324,…

4. Find the 8th term of the geometric sequence whosethird term is 1/25 and whosesixth term is 1/3125.

8 1

8

7

8

115

115

1

78125

a

a

Geometric Sequences & Series

The Sum of a Finite Geometric Sequence

The Sum of a Finite Geometric Sequence

The sum of the finite geometric sequence

a1, a1r, a1r2, a1r3, a1r4, …, a1rn–1

with common ratio r 1 is given by

11

1

1

1

nni

ni

a rS ar

r

Geometric Sequences & Series

4

1

2 1.5n

n

Examples:

Find the sum

41 2 3 4

1

4

4

1

2 1.5 2 1.5 2 1.5 2 1.5 2 1.5

3 1 1.5

1 1.53 4.0625

0.512.1875

0.5

2 1.5 24.375

n

n

n

n

Geometric Sequences & Series

How does one find the sum of an infinite geometric series?

The Sum of an Infinite Geometric Series

If |r| < 1, then the infinite geometric series

a1 + a1r + a1r2 + a1r3 +… + a1rn–1 + …

has the sum

11

0 1i

i

aS a r

r

Geometric Sequences & Series

1

1

4 0.5n

n

Example:

Find the sum of the infinite geometric series defined by

1

1

1

1

44 0.5

1 0.5

4

0.5

4 0.5 8

n

n

n

n

Remember to use theformula for an infinite

geometric series. Why?

Geometric Sequences & Series

Example:

Brandon deposits $100 at the beginning of each month into an account that pays 6% interest compounded monthly. What isthe balance at the end of 3 years?

1

36

36

.06100 1 100 1.005

12

100.5 1 1.005

1 1.005$3,953.28

a

S

Remember the firstdeposit will earn interest for 36 months,but the last deposit onlyearns interest for onemonth which is a1.

Geometric Sequences & Series

What you should know:

1. How to find the nth term of a geometric sequence.

2. How to find the partial sum of a geometric sequence (applying the sum formula for a finite geometric series).

3. How to find the sum of an infinite geometric series.