Copyright © Cengage Learning. All rights reserved. Sequences and Series Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 12.3 Geometric Sequences Copyright © Cengage Learning. All rights reserved.

Objectives Geometric Sequences Partial Sums of Geometric Sequences What Is an Infinite Series? Infinite Geometric Series

Geometric Sequences

Geometric Sequences An 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.

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.

Example 1 – Geometric Sequences cont’d (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.

Example 1 – Geometric Sequences cont’d (c) The sequence is a geometric sequence with a = 1 and r = The nth term is .

Example 1 – Geometric Sequences cont’d (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

Geometric Sequences We can find the nth term of a geometric sequence if we know any two terms, as the following example shows.

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.

Example 2 – Finding Terms of a Geometric Sequence Thus an = 5(3)n – 1 The eighth term is a8 = 5(3)8 – 1 = 5(3)7 = 10,935.

Partial Sums of Geometric Sequences

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 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

Partial Sums of Geometric Sequences So Sn(1 – r) = a(1 – rn) Sn = (r ≠ 1) We summarize this result.

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.

What Is an Infinite Series?

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. 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.

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).

Infinite Geometric Series

Infinite Geometric Series An 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 r n gets close to 0 as n gets large (you can easily convince yourself of this using a calculator).

Infinite Geometric Series It 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 ).

Example 6 – Infinite Series Determine 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

Example 6 – Solution cont’d (b) This is an infinite geometric series with a = 1 and r = . Since | r | = | | > 1, the series diverges.