Solution
= Solution
We prove <Cassini's identity> by <mathematical induction>. At $n=1$,
$$
F_0F_2-F_1^2=0\cdot1-1=-1=(-1)^1.
$$
Suppose $D_n=F_{n-1}F_{n+1}-F_n^2=(-1)^n$. The <Fibonacci recurrence> gives $F_{n-1}=F_{n+1}-F_n$ and hence
$$
\begin{aligned}
D_{n+1}
&=F_nF_{n+2}-F_{n+1}^2\\
&=F_n(F_{n+1}+F_n)-F_{n+1}^2\\
&=-\bigl((F_{n+1}-F_n)F_{n+1}-F_n^2\bigr)
=-D_n=(-1)^{n+1}.
\end{aligned}
$$
The induction proves
$$
\boxed{F_{n-1}F_{n+1}-F_n^2=(-1)^n\qquad(n\ge1).}
$$