Dattaprabhakar G. answered • 09/11/14
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Let n = 1. fn+1 fn-1 - fn 2 = f2 f0 - f1 2 = (1)(0) - (1)2 = (-1)1 so the identity holds for n =1
For n =2, fn+1 fn - 1 - fn2 = (2)(1) - (1)2 = 1 = (-1)2, so the identity holds. (Note: It is necessary to verify the identity for n = 1 AND n= 2 because of the term (-1)n)
For n+1, you must show
(fn+2)(fn) - (fn+1)2 = (-1)n+1
fn+2 = fn+1 + fn
(fn+1 + fn)(fn) = (fn+1)(fn) + (fn)2 = (fn+1)(fn) + [(fn+1)(fn-1) - (-1)n] by the Induction Hypothesis.
= (fn+1)[(fn) + (fn-1)] + (-1)n+1
= (fn+1)2 + (-1)n+1
Therefore, (fn+2)(fn) - (fn+1)2 = (-1)n+1 , which is what we wanted to show.
Post a comment. Thanks.
Dr. G.
