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College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson. Sequences and Series. 9. Geometric Sequences. 9.3. Introduction. Geometric sequences occur frequently in applications to fields such as: Finance Population growth. Geometric Sequences.
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College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson
Introduction • Geometric sequences occur frequently in applications to fields such as: • Finance • Population growth
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 multiplyby 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.
Example (a) E.g. 1—Geometric Sequences • 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 (b) E.g. 1—Geometric Sequences • 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 (c) E.g. 1—Geometric Sequences • The sequence • is a geometric sequence with a = 1 and r = ⅓ • The nth term is: an = (⅓)n–1
Example (d) E.g. 1—Geometric Sequences • 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
Example (d) E.g. 1—Geometric Sequences • 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 hnthat 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 = ar2a6 = 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:
E.g. 3—Finding Terms of a Geometric Sequence • Substituting for r in the first equation, 63/4 = ar2, gives: • It follows that the nth term of this sequence is: an = 7(3/2)n–1 • Thus, the fifth term is:
Partial Sums of Geometric Sequences • For the geometric sequence a, ar, ar2, ar3, ar4, . . . , arn–1, . . . , the nth partial sum is:
Partial Sums of Geometric Sequences • To find a formula for Sn, we multiply Snby r and subtract from Sn:
Partial Sums of Geometric Sequences • So, • 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:
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 Snwith n = 5, we get: • 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 • 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:
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: • This is an infinite series. • We note two things about it.
Infinite Series • It’s clear that, no matter how many terms of this series we add, the total will never exceed 1. • 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:
Infinite Series • To make this more precise, let’s look at the partial sums of this series:
Infinite Series • In general (see Example 5 of Section 9.1), • As n gets larger and larger, we are adding more and more of the terms of this series. • Intuitively, as n gets larger, Sngets 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, Sngets 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 Sngets close to a finite number S as n gets large, we say that: • S is the sum of the infinite 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: • It can be shown that, if |r| < 1, rngets close to 0 as n gets large. • You can easily convince yourself of this using a calculator.
Infinite Geometric Series • It follows that Sngets close to a/(1 – r) as n gets large, or • Thus, the sum of this infinite geometric series is: a/(1 – r)
Sum of an Infinite Geometric Series • If |r| < 1, the infinite geometric series a + ar + ar2 + ar3 + ar4 + . . . + arn–1 + . . . has the sum
E.g. 6—Finding the Sum of an Infinite Geometric Series • Find the sum of the infinite geometric series • We use the formula. • Here, a = 2 and r = (1/5). • So, the sum of this infinite series is:
E.g. 7—Writing a Repeated Decimal as a Fraction • Find the fraction that represents the rational number . • This repeating decimal can be written as a series:
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:
E.g. 7—Writing a Repeated Decimal as a Fraction • So, the sum of this part of the series is: • Thus,
