Question-76

Question-76

Let fn be the nth Fibonacci number. Prove that f_{1}^{2} + f_{2}^{2} +.....+ f_{n}^{2} = f_{n}f_{n+1} when n is a positive integer.

 

Solution

The basis step (n = 1) is clear, since f_{1}^{2} = f_{1}f_{2} = 1 . Assume the inductive hypothesis. Then f_{1}^{2}+f_{2}^{2}+....+f_{n}^{2}+f_{n+1}^{2} as desired.

= fnfn +1 + f_{n+1}^{2}

= fn+1(fn + fn+1)

=fn + 1 fn + 2 ,

The basis step (n = 1) is clear , since $f_{1}^{2 = f1f2 = 1. Assume the inducive

 

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