How do I prove $F_{n+1}^2 - F_nF_{n+2} = (-1)^n$ using induction? $F_n$ refers to the $n$ term of the Fibonacci Sequence.
I think I'm supposed to prove this by induction. I already have the base case. I am at:
$\text{F}_\text{k+1}^2 - F_k\text{F}_\text{k+2}  + \text{F}_\text{k+2}^2 - \text{F}_\text{k+1}\text{F}_\text{k+3} =  (-1)^k +\text{F}_\text{k+2}^2 - \text{F}_\text{k+1}\text{F}_\text{k+3}$
So it's probably easier to work backwards. Multiply $-1$ in order to get $(-1)^\text{k + 1}$  and then get:
$-\text{F}_\text{k+1}^2 + F_k\text{F}_\text{k+2}  - \text{F}_\text{k+2}^2 + \text{F}_\text{k+1}\text{F}_\text{k+3} =  (-1)^\text{k+1} -\text{F}_\text{k+2}^2 + \text{F}_\text{k+1}\text{F}_\text{k+3}$
So then bring everything to the left side and mess around with the left side. 
$-\text{F}_\text{k+1}^2 + F_k\text{F}_\text{k+2}  - \text{F}_\text{k+2}^2 + \text{F}_\text{k+1}\text{F}_\text{k+3} +\text{F}_\text{k+2}^2 - \text{F}_\text{k+1}\text{F}_\text{k+3} =  (-1)^\text{k + 1} $
And so somehow get this to look more like this:
$\text{F}_\text{k+1}^2 - F_k\text{F}_\text{k+2}  + \text{F}_\text{k+2}^2 - \text{F}_\text{k+1}\text{F}_\text{k+3}$
...except I don't know how to do that. 
Thanks

Hint
\begin{align}
F_{n+1}^2-F_nF_{n+2} &= F_{n+1}^2-(F_{n+2}-F_{n+1})F_{n+2}=F_{n+1}^2-F_{n+2}^2+F_{n+1}F_{n+2} \\ &=F_{n+1}(F_{n+1}+F_{n+2})-F_{n+2}^2=F_{n+1}F_{n+3}-F_{n+2}^2.
\end{align}
