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(c) The Fibonacci sequence h fn j is defined recursively by the conditions f1 − 1 f2 − 1 fn − fn21 1 fn22 n > 3 Each term is the sum of the two preceding terms. The first few terms are h1, 1, 2, 3, 5, 8, 13, 21, . . .j This sequence arose when the 13th-century Italian mathematician known as Fibonacci solved a problem concerning the breeding of rabbits (see Exercise 83). n A sequence such as the one in Example 1(a), an − nysn 1 1d, can be pictured either by plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2. Note that, since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinates s1, a1d s2, a2 d s3, a3 d . . . sn, an d . . . From Figure 1 or Figure 2 it appears that the terms of the sequence an − nysn 1 1d are approaching 1 as n becomes large. In fact, the difference 1 2 n n 1 1 − 1 n 1 1 can be made as small as we like by taking n sufficiently large. We indicate this by writing lim nl` n n 1 1 − 1 In general, the notation lim nl` an − L means that the terms of the sequence han j approach L as n becomes large. Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity given in Section 2.6.
by: PIZZAspartan
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