1062 CHAPTER 11 Further Topics in Algebra EXAMPLE 3 FindingTerms of a Geometric Sequence Determine r and a1 for the geometric sequence with a3 = 20 and a6 = 160. SOLUTION We obtain a6 by multiplying a3 by the common ratio three times. a6 = a3r 3 Definition of geometric sequence 160 = 20r3 Let a 6 = 160 and a3 = 20. 8 = r3 Divide by 20. r = 2 Take cube roots and interchange sides. Now use this value of r and the fact that a3 = 20 to find the first term, a1. an = a1r n-1 Formula for a n 20 = a11223-1 Let a n = 20, r = 2, and n = 3. 20 = a1142 Apply the exponent. a1 = 5 Divide by 4 and interchange sides. S Now Try Exercise 27. EXAMPLE 2 FindingTerms of a Geometric Sequence Determine a5 and an for the geometric sequence 4, 12, 36, 108, c . SOLUTION The first term, a1, is 4. Find r by choosing any term after the first and dividing it by the preceding term. For example, r = 36 12 = 3. a5 = a4 # r Definition of geometric sequence a5 = 108 # 3 Let a4 = 108 and r = 3. a5 = 324 Multiply. The nth term is found as follows. an = a1r n-1 Formula for a n an = 4132n-1 Let a 1 = 4 and r = 3. We can also find the fifth term by replacing n with 5 in this formula. a5 = 41325-1 = 41324 = 324 S Now Try Exercise 21. EXAMPLE 4 Modeling a Population of Fruit Flies A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. Suppose there are 100 insects in the first generation. How many would there be in the fourth generation? Round to the nearest whole number. SOLUTION Consider the list of populations as a geometric sequence with a1 as the first-generation population, a2 the second-generation population, and so on. Then the fourth-generation population is a4. an = a1r n-1 Formula for a n a4 = 10011.524-1 Let a 1 = 100, r = 1.5, and n = 4. a4 = 10013.3752 Apply the exponent. a4 = 338 Multiply. Round to the nearest whole number. In the fourth generation, the population will number 338 insects. S Now Try Exercise 73.
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