copyright © cengage learning. all rights reserved. 8.3 geometric sequences and series

Copyright © Cengage Learning. All rights reserved.

8.3 Geometric Sequences and Series

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What You Should Learn

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

• Find nth partial sums of geometric sequences.

• Find sums of infinite geometric series.

• Use geometric sequences to model and solve real-life problems.

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Geometric Sequences

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Geometric Sequences

We know that a sequence whose consecutive terms have a common difference is an arithmetic sequence.

In this section, you will study another important type of sequence called a geometric sequence. Consecutive terms of a geometric sequence have a common ratio.

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Geometric Sequences

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Example 1 – Examples of Geometric Sequences

a. The sequence whose nth term is 2n is geometric. For this

sequence, the common ratio between consecutive terms

is 2.

2, 4, 8, 16, . . . , 2n, . . .

b. The sequence whose nth term is 4(3n) is geometric. For this sequence, the common ratio between consecutive terms is 3.

12, 36, 108, 324, . . . , 4(3n), . . .

Begin with n = 1.

Begin with n = 1.

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Example 1 – Examples of Geometric Sequences

c. The sequence whose nth term is is geometric. For this sequence, the common ratio between consecutive terms is

Begin with n = 1.

cont’d

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Geometric Sequences

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The Sum of a Finite Geometric Sequence

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The Sum of a Finite Geometric Sequence

The formula for the sum of a finite geometric sequence is as follows.

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Example 6 – Finding the Sum of a Finite Geometric Sequence

Find the following sum.

Solution:

By writing out a few terms, you have

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Example 6 – Solution

Now, because

a1 = 4(0.3), r = 0.3, and n = 12

you can apply the formula for the sum of a finite geometric sequence to obtain

1.71.

cont’d

Formula for sum of a finite geometric sequence

Substitute 4(0.3) for a1, 0.3 for r, and 12 for n.

Use a calculator.

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Geometric Series

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Geometric Series

The sum of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series.

The formula for the sum of a finite geometric sequence can, depending on the value of r, be extended to produce a formula for the sum of an infinite geometric series.

Specifically, if the common ratio has the property that

| r | < 1

then it can be shown that r

n becomes arbitrarily close to zero as n increases without bound.

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Geometric Series

Consequently,

This result is summarized as follows.

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Example 7 – Finding the Sum of an Infinite Geometric Series

Find each sum.

a.

b. 3 + 0.3 + 0.03 + 0.003 + . . .

Solution:

a.

= 10

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Example 7 – Solution

b. 3 + 0.3 + 0.03 + 0.003 + . . .

= 3 + 3(0.1) + 3(0.1)2 + 3(0.2)3 + . . .

3.33

cont’d