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## Homework Statement

The fibonacci numbers are defined by F

_{0}= 0 F

_{1}= 1 and F

_{n}=

_{n-1}+ F

_{n-2}for n >= 2.

Use induction the prove the following:

F

_{n-1}F

_{n+1}- F

_{n}= (-1)

^{n}

**The attempt at a solution**

Let P(n) = F

_{n-1}F

_{n+1}- F

_{n}

^{2}= (-1)

^{n}where n>= 1

Show it holds for first natural number:

P(1) = F

_{0}+ F

_{2}- F

_{1}

^{2}= -(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)-1}F

_{(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.