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Math 104: Homework 1-
1. The Fibonacci numbers are defined by F0 = 0 and F1 = 1, and
Fn+1 = Fn + Fn-1
for n ∈ N. Let the golden ratio be defined as ? = (1+√5/2).
(a) Show that ?2 = 1 + ?.
(b) Let f(n) = (?n - (1 - ?)n/√5).
For n ∈ N, define Hn to be the hypothesis that "both Fn = f(n) and Fn-1 = f(n - 1)". Apply mathematical induction to prove that Hn is true for all n ∈ N, and deduce that Fn = f(n) for all n ∈ N ∪ {0}. [Hint: it is simpler to carry out the algebra in terms of ? and use the identity in (a), as opposed to calculating explicitly in terms of (1 + √5)/2.].
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