1Copyright © Cengage Learning. All rights reserved.

12.3 Geometric Sequences

2

Objectives

► Geometric Sequences

► Partial Sums of Geometric Sequences

► What Is an Infinite Series?

► Infinite Geometric Series

3

Geometric Sequences

4

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.

5

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.

6

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

7

Example 1 – Geometric Sequences

(c) The sequence

is a geometric sequence with a = 1 and r =

The nth term is .

cont’d

8

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

9

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

10

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.

11

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.

12

Partial Sums of Geometric Sequences

13

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

14

Partial Sums of Geometric Sequences

So Sn(1 – r) = a(1 – rn)

Sn = (r ≠ 1)

We summarize this result.

15

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.

16

What Is an Infinite Series?

17

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.

18

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).

19

Infinite Geometric Series

20

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).

21

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).

22

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

23

Example 6 – Solution(b) This is an infinite geometric series with a = 1 and r = .

Since |r | = | | > 1, the series diverges.

cont’d