1042 CHAPTER 11 Further Topics in Algebra Leonardo of Pisa (Fibonacci) (1170–1250) One of the most famous sequences in mathematics is the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, c, named for the Italian mathematician Leonardo of Pisa, who was also known as Fibonacci. The Fibonacci sequence may be defined using a recursion formula. See Exercise 33. Figure 4 EXAMPLE 2 Using a Recursion Formula Find the first four terms of each sequence. (a) a1 = 4 an = 2 # an-1 + 1, if n 71 (b) a1 = 2 an = an-1 + n - 1, if n 71 SOLUTION (a) This is a recursive definition. We know that a1 = 4. Use an = 2 # an-1 + 1 to find subsequent terms. a1 = 4 a2 = 2 # a1 + 1 = 2 # 4 + 1 = 9 a3 = 2 # a2 + 1 = 2 # 9 + 1 = 19 a4 = 2 # a3 + 1 = 2 # 19 + 1 = 39 (b) In this recursive definition, a1 = 2 and an = an-1 + n - 1. a1 = 2 a2 = a1 + 2 - 1 = 2 + 1 = 3 a3 = a2 + 3 - 1 = 3 + 2 = 5 a4 = a3 + 4 - 1 = 5 + 3 = 8 S Now Try Exercises 31 and 35. EXAMPLE 3 Modeling Insect Population Growth The population of an insect often grows rapidly at first, then levels off because of competition for limited resources. In one study, the behavior of the winter moth was modeled with a sequence similar to the following, where an represents the population density, in thousands per acre, during year n. (Data from Varley, G., and G. Gradwell, “Population models for the winter moth,” Symposium of the Royal Entomological Society of London.) a1 = 1 an = 2.85an-1 - 0.19an-1 2, for n Ú 2 (a) Give a table of values for n = 1, 2, 3, c , 10. (b) Graph the sequence. Describe what happens to the population density. SOLUTION (a) Evaluate a1, a2, a3, c , a10 recursively. We are given a1 = 1. a2 = 2.85a1 - 0.19a1 2 = 2.85112 - 0.191122 = 2.66 a3 = 2.85a2 - 0.19a2 2 = 2.8512.662 - 0.1912.6622 ≈6.24 Approximate values for an are shown in the table. Figure 4 shows computation of the sequence, denoted by u1n2 rather than an, using a calculator. n 12345678910 an 1 2.66 6.24 10.4 9.11 10.2 9.31 10.1 9.43 9.98
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