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1Copyright © Cengage Learning. All rights reserved.
12.3 Geometric Sequences
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Objectives
► Geometric Sequences
► Partial Sums of Geometric Sequences
► What Is an Infinite Series?
► Infinite Geometric Series
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Geometric Sequences
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Geometric SequencesAn arithmetic sequence is generated when we repeatedly add a number d to an initial term a. A geometric sequence is generated when we start with a number a and repeatedly multiply by a fixed nonzero constant r.
The number r is called the common ratio because the ratio of any two consecutive terms of the sequence is r.
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Example 1 – Geometric Sequences
(a) If a = 3 and r = 2, then we have the geometric sequence
3, 3 2, 3 22, 3 23, 3 24, . . .
or
3, 6, 12, 24, 48, . . .
Notice that the ratio of any two consecutive terms is r = 2.
The nth term is an = 3(2)n – 1.
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Example 1 – Geometric Sequences
(b) The sequence
2, –10, 50, –250, 1250, . . .
is a geometric sequence with a = 2 and r = –5.
When r is negative, the terms of the sequence alternate in sign.
The nth term is an = 2(–5)n – 1.
cont’d
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Example 1 – Geometric Sequences
(c) The sequence
is a geometric sequence with a = 1 and r =
The nth term is .
cont’d
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Example 1 – Geometric Sequences
(d) The graph of the geometric sequence is shown in Figure 1.
Notice that the points in the graph lie on the graph of the exponential function y = .
If 0 < r < 1, then the terms of the geometric sequence arn – 1 decrease, but if r > 1, then the terms increase.
Figure 1
cont’d
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Geometric SequencesWe can find the nth term of a geometric sequence if we know any two terms, as the following example shows.
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Example 2 – Finding Terms of a Geometric Sequence
Find the eighth term of the geometric sequence 5, 15, 45, . . . .
Solution:To find a formula for the nth term of this sequence, we need to find a and r. Clearly, a = 5.
To find r, we find the ratio of any two consecutive terms.
For instance, r = = 3.
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Example 2 – Finding Terms of a Geometric Sequence
Thusan = 5(3)n – 1
The eighth term is
a8 = 5(3)8 – 1
= 5(3)7
= 10,935.
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Partial Sums of Geometric Sequences
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Partial Sums of Geometric Sequences
For the geometric sequence a, ar, ar2, ar3, ar4, . . . , ar n – 1, . . . , the nth partial sum is
Sn = = a + ar + ar2 + ar3 + ar4 + · · · + ar n – 1
To find a convenient formula for Sn, we multiply Sn by r and subtract from Sn.
Sn = a + ar + ar2 + ar3 + ar4 + · · · + ar n – 1
rSn = ar + ar2 + ar3 + ar4 + · · · + ar n – 1 + arn
Sn – rSn = a – arn
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Partial Sums of Geometric Sequences
So Sn(1 – r) = a(1 – rn)
Sn = (r ≠ 1)
We summarize this result.
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Example 4 – Finding a Partial Sum of a Geometric Sequence
Find the sum of the first five terms of the geometric sequence 1, 0.7, 0.49, 0.343, . . .
Solution:The required sum is the sum of the first five terms of a geometric sequence with a = 1 and r = 0.7. Using the formula for Sn with n = 5, we get
Thus the sum of the first five terms of this sequence is 2.7731.
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What Is an Infinite Series?
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What Is an Infinite Series?
An expression of the form
= a1 + a2 + a3 + a4 + . . .
is called an infinite series.The dots mean that we are to continue the addition indefinitely. Let an = 1/2n. As we have seen in Section 12.1, then the nth partial sum
Sn = 1- 1/2n
On the other hand, as n gets larger and larger, we are adding more and more of the terms of this series. Intuitively, as n gets larger, Sn gets closer to the sum of the series.
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What Is an Infinite Series?Now, notice that as n gets large, 1/2n gets closer and closer to 0.
Thus Sn gets close to 1 – 0 = 1. We can write
Sn 1 as n
In general, if Sn gets close to a finite number S as n gets large, we say that the infinite series converges (or is convergent).
The number S is called the sum of the infinite series. If an infinite series does not converge, we say that the series diverges (or is divergent).
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Infinite Geometric Series
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Infinite Geometric SeriesAn infinite geometric series is a series of the form
a + ar + ar2 + ar3 + ar4 + . . . + ar n – 1 + . . .
We can apply the reasoning used earlier to find the sum of an infinite geometric series. The nth partial sum of such a series is given by the formula
(r ≠ 1)
It can be shown that if |r | < 1, then rn gets close to 0 as n gets large (you can easily convince yourself of this using a calculator).
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Infinite Geometric SeriesIt follows that Sn gets close to a/(1 – r) as n gets large, or
Sn as n
Thus the sum of this infinite geometric series is a/(1 – r).
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Example 6 – Infinite SeriesDetermine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum.
(a) (b)
Solution:(a) This is an infinite geometric series with a = 2 and r = .
Since |r | = | | < 1, the series converges. By the formula for the sum of an infinite geometric series we have
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Example 6 – Solution(b) This is an infinite geometric series with a = 1 and r = .
Since |r | = | | > 1, the series diverges.
cont’d
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