RabbitRedux

# RabbitRedux - i th term in the sequence is usually denoted...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Rabbit Redux Many of you are probably familiar with the famous sequence of numbers ﬁrst devised by Leonardo Fibonacci (who lived from c. 1170 to c. 1250), and named in his honor: 1. The ﬁrst two terms in the sequence are 1, 1. 2. Thereafter, each successive term is found by adding the previous two. That is, the third term is 1 + 1 = 2, the fourth term is 1 + 2 = 3, the ﬁfth is 2 + 3 = 5, and so on. Thus far we have 1 , 1 , 2 , 3 , 5 , ... . Fibonacci’s sequence comes up so frequently in mathematics that there’s even a research journal (called the Fibonacci Quarterly ) dedicated only to papers related to the sequence! Fibonacci himself had in mind using this sequence to approximate the growth of a population of rabbits. Why might this be? Investigate the problem by answering the questions given below on your own paper. 1. Compute the ﬁrst 20 terms in the Fibonacci sequence. (The
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Unformatted text preview: i th term in the sequence is usually denoted F i , so F 1 = 1, F 2 = 1, F 2 = 2, and so forth. This notation might help you write your answer more clearly!) 2. Graph (roughly!) the terms you’ve computed on a set of axes, where the x-axis corresponds to the index of a given term in the sequence, and the y-axis corresponds to the value of the term. For instance, the ﬁrst term, F 1 = 1, would be plotted at the point (1 , 1); the second term, F 2 = 1, would be plotted at the point (2 , 1), the third term, F 3 = 2, at (3 , 2), and so forth. 3. Which of the models we’ve investigated does the Fibonacci sequence most closely resemble? Why might such a model be a somewhat realistic one to describe the growth of a rabbit population? Please explain your answers carefully....
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