I can't seem to solve this problem. It is:

> 
> The Fibonacci numbers $F(0), F(1), F(2),\dots $ are defined as follows:
> 
> 
> \begin{align}
> F(0) &::= 0 \\
> F(1) &::= 1 \\
> F(n) &::= F(n-1) + F(n-2)\qquad(\forall n \ge 2)\end{align}
> 
> 
> Thus, the first Fibonacci numbers are $0, 1, 1, 2, 3, 5, 8, 13,$ and $21$. Prove by induction that $\forall n \ge1$,
> 
> 
> $$F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$$
> 
> 
> 

I'm stuck, as I my induction hypothesis was the final equation, and I replaced n in it with n+1, which gave me:

$$F(n) \cdot F(n+2) - F(n+1)^2 = (-1)^{n+1}$$

I then tried simplifying this using the first equation, which gave me:
$$[(F(n-1) + F(n-2)]\cdot F(n+2) - F(n+1)^2 = (-1)^{n+1}$$

I then tried replacing $n$ in the first equation with $n+1$, but that just gave me 

$$2F(n-1) + F(n-2)$$

I'm really not sure how to proceed, and I was hoping for some help. I'm new to induction and I'm hoping this is just an algebra problem and not a problem with the method, but any help would be greatly appreciated.