Consider the ratio between two consecutive Fibonacci numbers fn for n = 2, 3, .... I'n fn-1 (a) Show that the sequence of ratios (rn)nEN satisfies the recursive relation 1 T'n = 1+ T'n-1 Assume that this sequence converges.' Show that the limit ri is a fixed point of the function f(r) = 1+, i.e., f(ri) = r1. Show that r is precisely the 'golden ratio', ry (1+ V5). (b) lim,+0 "'n (c)

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.