Copyright © Cengage Learning. All rights reserved. 8. 3 Geometric Sequences and Series
Dec 24, 2015
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