9.3 Geometric Sequences and Series

Objective

• To find specified terms and the common ratio in a geometric sequence.

• To find the partial sum of a geometric series

Geometric Sequences

• Consecutive terms of a geometric sequence have a common ratio.

Definition of a Geometric Sequence

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

• The number r is the common ratio of the sequence.

32 4

1 2 3

, , ,..., 0aa a

r r r ra a a

Example 1Examples of Geometric

Sequences• a). The sequence whose nth term is

• b). The sequence whose nth term is

• C) The sequence whose nth term is

2n

4(3 )n

1( )

3n

• Notice that each of the geometric sequences has an nth term that is of the form where the common ratio is r.

• A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers.

nar

The nth Term of a Geometric Sequence

• The nth term of a geometric sequence has the form where r is the common ratio of consecutive terms of the sequence.

11n

na a r

• So, every geometric sequence can be written in the following form,

1 2 3 4

2 3 11 1 1 1 1

,. . .

, , , ,

n

n

a a a a a

a a r a r a r a r

• If you know the nth term of a geometric sequence, you can find the (n+1)th term by multiplying by r. that is

1n na ra

Example 2Finding the Terms of a Geometric Sequence

Write the first five terms of the geometric sequence whose first term is

and whose common ratio is r = 2.

3, 6, 12, 24, 48

1 3a

Example 3Finding a Term of a

Geometric Sequence• Find the 15th term of the geometric sequence

whose first term is 20 and whose common ration is 1.05.

1415 20(1.05) 39.6a

Example 4Finding a Term of a Geometric

Sequence• Find the 12th term of the geometric

sequence 5, 15, 45, . . .

1112

3

5(3) 885735

r

a

• If you know any two terms of a geometric sequence, you can use that information to find a formula for the nth term of the sequence.

Example 5Finding a Term of a

Geometric Sequence

• The fourth term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. (assume that the terms of the sequence are positive.)

34 1

910 1

1

1314

125

125

64Two equations in two unknowns

Solve using substitution or elimination

1, 1000

21

1000( ) .122072

a a r

a a r

r a

a

The Sum of a Finite Geometric Sequence

• The sum of the geometric sequence

with common ratio is given by

2 3 11 1 1 1 1, , , , na a r a r a r a r

1r

11 1

1

1

1

nni

ni

rS a r a

r

Example 6Finding the Sum of a Finite

Geometric Sequence• Find the sum

121

1

4(0.3)i

i

1

12

12 1

12

4, .3, 12

(1 )

(1 )

(1 (.3) )4

(1 .3)

5.71

a r n

rS a

r

• When using the formula for the sum of a finite geometric sequence, be careful to check that the index begins at . If the index begins at , you must adjust the formula for the th partial sum.

4

0

4(0.3)i

i

These are not the same, be careful of the indices

40 1 2 3 4

0

41 1 2 3 4

1

4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3)

4(0.3) 4(0.3) 4(0.3) 4(0.3) 4(0.3)

i

i

i

i

Geometric Series

• The summation of the terms of an infinite geometric sequence is called an infinite geometric series or geometric series.

The sum of an Infinite Geometric Series

• If the infinite geometric series

has the sum

1,r 2 3 1

1 1 1 1 1, , , , ,...na a r a r a r a r

11

0 1i

i

aS a r

r

Example 7Finding the Sums of an

Infinite Geometric Series

• Find the sums.

• a)

• b)

0

4(0.6)n

n

3 0.3 0.03 0.003 ...

0

44(0.6) 10

1 .6

33 0.3 0.03 0.003 ...

1 .13 1

3.9 3

n

n

ApplicationsCompound Interest

• A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance of this annuity at the end of 2 years?

2424

2323

11

24

24

24

.0650(1 )

12.06

50(1 )12

.

.

.

.0650(1 )

12

1 (1.005)50(1 .005)

1 (1.005)

$1277.95

A

A

A

S

S