CMPSCI 250 Discussion #5: Fibonacci Numbers

David Mix Barrington

17 October 2007

In this Discussion we study the Fibonacci numbers, first described by Leonardo of Pisa in the 1200's. The n'th Fibonacci number F(n) is formally defined by the following recursive rules:

For example, let S(n) be the sum of F(i) as i goes from 1 to n. A look at examples suggests that S(n) = F(n+2) - 1. Since S(n+1) = S(n) + F(n+1) by definition, we can apply the inductive hypothesis to S(n) to get S(n+1) = F(n+2) - 1 + F(n+1) which is F(n+3) - 1 by the Fibonacci definition, whch is F((n+1)+2) - 1 as required. The base case is n=0, where we need only check that S(0) = 0 = F(2) - 1, which is true because F(1) = 1.

Proofs of facts about the Fibonacci numbers often use strong induction. The truth of the statement P(n+1) normally involves both P(n) and P(n-1) because F(n+1) is defined in terms of both F(n) and F(n-1). We need two base cases to get started, typically P(0) and P(1), since otherwise we would not be able to prove the second case from the inductive argument without two prior cases to use.

Writing Exercise: Write a careful inductive proof, for all naturals n after an appropriate starting point, of the two following statements:

Last modified 18 October 2007