By calculating for $n\in \{1,2,3,4,5,6,7\}$, I've formulated the rule \begin{equation} F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n, \end{equation} where $F_n$ is the $n$th fibonacci number. I want to show that this is true for all $n \in \mathbb{N}$.

I tried using induction, with $n=1$ as the basis step, but didn't get very far:

For the induction step, we assume the formula holds for a $n = k$, and checks for $n=k+1$: \begin{align*} F_{k} \cdot F_{k+2} &= F_{k} \cdot (F_{k+1} + F_{k}) \\ &= F_k \cdot F_{k+1} + F_k^2 \\ \end{align*} If somehow $F_k \cdot F_{k+1} = (-1)^k$, then I would be done. But I don't see how that's possible.

Is there a better way of proving this, maybe without using induction? Or am I just going about it the wrong way?