The fibonacci numbers are defined by F0 = 0 F1 = 1 and Fn = n-1 + Fn-2 for n >= 2.
Use induction the prove the following:
Fn-1Fn+1 - Fn = (-1)n
The attempt at a solution
Let P(n) = Fn-1Fn+1 - Fn2 = (-1)n where n>= 1
Show it holds for first natural number:
P(1) = F0 + F2 - F12 = -(1) = -1. So it is true
Now assume it works for some k and prove it works for k+1
This is where I'm really not sure what to do. I have tried putting
F(k+1)-1F(k+1)+1 - F(k+1)2 but it ultimately gets me no where.
Usually when I have done inductive proofs I start with some sequence of numbers 1,2,3,4.....,n and show n+1 is there but this one seems slightly different and I'm not sure how to proceed.
Any help is appreciated.