I am trying to figure out how to show that $\displaystyle f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^{n}$ when n is a positive integer.

My Work

P(n) is $\displaystyle f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^n$ for $\displaystyle n>=1$

Basis step - P(1) is true because $\displaystyle f_{2}f_{0}-f_{1}^{2}=1*0-1=-1=(-1)^1$

Inductive step - Assume $\displaystyle P(k)=f_{k+1}f_{k-1}-f_{k}^{2}=(-1)^k$ is true,

then $\displaystyle P(k+1)=f_{k+2}f_{k}-(f_{k+1}^2=(-1)^{k+1} $is true

Then prove $\displaystyle P(k+1)=[f_{k+1}f_{k-1}-f_{k}^{2}]+[f_{k+2}f_{k}-f_{k+1}^{2}]=(-1)^k+[f_{k+2}f_{k}-f_{k+1}^2]$, but I can't figure out how to get this P(k+1) equation to equal $\displaystyle (-1)^{k+1} $

I received the reply below, which is probably a wonderful answer, but I am unclear as to where in the 3rd line came from. I though you had to add the initial part of the P(k+1) equation, which is $\displaystyle f_{k+1}f_{k}-f_{k+1}^2$, to both sides of the P(k) equation as I did above.

First Reply

Assume for n.

We want to show that

,<--- negative of the induction hypothesis

by the induction hypothesis.

So, is true.