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College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson 9 Sequences and Series 9.3 Geometric Sequences Introduction Geometric sequences occur frequently in applications to fields such as: • Finance • Population growth Geometric Sequences Arithmetic Sequence vs. Geometric Sequence 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. Geometric Sequence—Definition A geometric sequence is a sequence of the form a, ar, ar2, ar3, ar4, . . . • The number a is the first term. • r is the common ratio of the sequence. • The nth term of a geometric sequence is given by: an = arn–1 Common Ratio The number r is called the common ratio because: • The ratio of any two consecutive terms of the sequence is r. E.g. 1—Geometric Sequences Example (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 E.g. 1—Geometric Sequences Example (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. E.g. 1—Geometric Sequences Example (c) The sequence 1 1 1 1 1, , , , ,... 3 9 27 81 is a geometric sequence with a = 1 and r = ⅓ • The nth term is: an = (⅓)n–1 E.g. 1—Geometric Sequences Example (d) Here’s the graph of the geometric sequence an = (1/5) · 2n – 1 • Notice that the points in the graph lie on the graph of the exponential function y = (1/5) · 2x–1 E.g. 1—Geometric Sequences Example (d) If 0 < r < 1, then the terms of the geometric sequence arn–1 decrease. However, if r > 1, then the terms increase. • What happens if r = 1? Geometric Sequences in Nature Geometric sequences occur naturally. Here is a simple example. • Suppose a ball has elasticity such that, when it is dropped, it bounces up one-third of the distance it has fallen. Geometric Sequences in Nature If the ball is dropped from a height of 2 m, it bounces up to a height of 2(⅓) = ⅔ m. • On its second bounce, it returns to a height of (⅔)(⅓) = (2/9)m, and so on. Geometric Sequences in Nature Thus, the height hn that the ball reaches on its nth bounce is given by the geometric sequence hn = ⅔(⅓)n–1 = 2(⅓)n • We can find the nth term of a geometric sequence if we know any two terms—as the following examples show. E.g. 2—Finding Terms of a Geometric Sequence Find the eighth term of the geometric sequence 5, 15, 45, . . . . • To find a formula for the nth term of this sequence, we need to find a and r. • Clearly, a = 5. E.g. 2—Finding Terms of a Geometric Sequence To find r, we find the ratio of any two consecutive terms. • For instance, r = (45/15) = 3 • Thus, an = 5(3)n–1 • The eighth term is: a8 = 5(3)8–1 = 5(3)7 = 10,935 E.g. 3—Finding Terms of a Geometric Sequence The third term of a geometric sequence is 63/4, and the sixth term is 1701/32. Find the fifth term. • Since this sequence is geometric, its nth term is given by the formula an = arn–1. • Thus, a3 = ar3–1 = ar2 a6 = ar6–1 = ar5 E.g. 3—Finding Terms of a Geometric Sequence From the values we are given for those two terms, we get this system of equations: • We solve this by dividing: ar 5 2 ar 1701 32 63 4 r3 27 8 r 3 2 634 ar 2 1701 5 32 ar E.g. 3—Finding Terms of a Geometric Sequence Substituting for r in the first equation, 63/4 = ar2, gives: 63 3 2 4 a 2 a7 • It follows that the nth term of this sequence is: an = 7(3/2)n–1 • Thus, the fifth term is: a5 7 3 51 2 7 3 4 2 567 16 Partial Sums of Geometric Sequences Partial Sums of Geometric Sequences For the geometric sequence a, ar, ar2, ar3, ar4, . . . , arn–1, . . . , the nth partial sum is: n Sn ar k 1 k 1 a ar ar ar ar ar 2 3 4 n 1 Partial Sums of Geometric Sequences To find a formula for Sn, we multiply Sn by r and subtract from Sn: Sn a ar ar ar ar ar 2 rSn 3 4 n 1 ar ar 2 ar 3 ar 4 ar n 1 ar n Sn rSn a ar n Partial Sums of Geometric Sequences So, a 1 r r 1 1 r Sn 1 r a 1 r n n Sn • We summarize this result as follows. Partial Sums of a Geometric Sequence For the geometric sequence an = arn–1, the nth partial sum Sn = a + ar + ar2 + ar3 + ar4 + . . . + arn–1 (r ≠ 1) is given by: n 1 r Sn a 1 r E.g. 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, . . . • The required sum is the sum of the first five terms of a geometric sequence with a = 1 and r = 0.7 E.g. 4—Finding a Partial Sum of a Geometric Sequence Using the formula for Sn with n = 5, we get: S5 1 1 0.7 1 0.7 5 2.7731 • The sum of the first five terms of the sequence is 2.7731. E.g. 5—Finding a Partial Sum of a Geometric Sequence Find the sum 5 7 k 1 2 3 k • The sum is the fifth partial sum of a geometric sequence with first term a = 7(–⅔) = –14/3 and common ratio r = –⅔. • Thus, by the formula for Sn, we have: 32 14 1 14 1 243 770 S5 5 2 3 1 3 3 243 3 2 3 5 What Is an Infinite Series? 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. Infinite Series What meaning can we attach to the sum of infinitely many numbers? • It seems at first that it is not possible to add infinitely many numbers and arrive at a finite number. • However, consider the following problem. Infinite Series You have a cake and you want to eat it by: • First eating half the cake. • Then eating half of what remains. • Then again eating half of what remains. Infinite Series This process can continue indefinitely because, at each stage, some of the cake remains. • Does this mean that it’s impossible to eat all of the cake? • Of course not. Infinite Series Let’s write down what you have eaten from this cake: 1 1 1 1 1 n 2 4 8 16 2 • This is an infinite series. • We note two things about it. Infinite Series 1. It’s clear that, no matter how many terms of this series we add, the total will never exceed 1. 2. The more terms of this series we add, the closer the sum is to 1. Infinite Series This suggests that the number 1 can be written as the sum of infinitely many smaller numbers: 1 1 1 1 1 1 n 2 4 8 16 2 Infinite Series To make this more precise, let’s look at the partial sums of this series: 1 S1 2 1 1 S2 2 4 1 1 1 S3 2 4 8 1 1 1 1 S4 2 4 8 16 1 2 3 4 7 8 15 16 Infinite Series In general (see Example 5 of Section 9.1), 1 Sn 1 n 2 • 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. 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. • Using the notation of Section 4.6, we can write: Sn → 1 as n → ∞ Sum of Infinite Series In general, if Sn gets close to a finite number S as n gets large, we say that: • S is the sum of the infinite series. Infinite Geometric Series Infinite Geometric Series An infinite geometric series is a series of the form a + ar + ar2 + ar3 + ar4 + . . . + arn–1 + . . . • We can apply the reasoning used earlier to find the sum of an infinite geometric series. Infinite Geometric Series The nth partial sum of such a series is given by: 1 r Sn a 1 r n r 1 • It can be shown that, if | r | < 1, rn 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 a Sn 1 r as • Thus, the sum of this infinite geometric series is: a/(1 – r) n Sum of an Infinite Geometric Series If | r | < 1, the infinite geometric series a + ar + ar2 + ar3 + ar4 + . . . + arn–1 + . . . has the sum a S 1 r E.g. 6—Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series 2 2 2 2 2 n 5 25 125 5 • We use the formula. • Here, a = 2 and r = (1/5). • So, the sum of this infinite series is: 2 5 S 1 1 5 2 E.g. 7—Writing a Repeated Decimal as a Fraction Find the fraction that represents the rational number 2.351. • This repeating decimal can be written as a series: 23 51 51 51 51 10 1000 100,000 10,000,000 1,000,000,000 E.g. 7—Writing a Repeated Decimal as a Fraction After the first term, the terms of the series form an infinite geometric series with: 51 a and 1000 1 r 100 E.g. 7—Writing a Repeated Decimal as a Fraction So, the sum of this part of the series is: S 51 1000 1 100 1 51 1000 99 100 51 100 51 1000 99 990 • Thus, 23 51 2328 388 2.351 10 990 990 165