Prove the given property of the Fibonacci numbers I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular:
For all $n \in \mathbb N$, $F_{n+1}  F_{n-1} - F_n^2 = (-1)^n$.
I tried to use induction but at some point on induction step I just loose the point. Any ideas?

Induction will work fine. Assuming it works for $n$, we have $$F_{n+2}F_n-F_{n+1}^2=(F_{n+1}+F_n)F_n-F_{n+1}^2=F_{n+1}(F_n-F_{n+1})+F_n^2=F_{n+1}(-F_{n-1})+F_n^2$$ $$=-\left(F_{n+1}F_{n-1}-F_n^2\right)=-(-1)^n=(-1)^{n+1}$$
