Solution
= Solution
For $n=0$, the difference is $F_1^2-F_0F_2=1=(-1)^0$. Define $D_n=F_{n+1}^2-F_nF_{n+2}$. The <Fibonacci recurrence> gives
$$
\begin{aligned}
D_{n+1}
&=F_{n+2}^2-F_{n+1}F_{n+3}\\
&=F_{n+2}(F_{n+2}-F_{n+1})-F_{n+1}^2\\
&=F_nF_{n+2}-F_{n+1}^2=-D_n.
\end{aligned}
$$
The <inductive step> therefore changes $(-1)^n$ into $(-1)^{n+1}$. By <mathematical induction>, \b[$D_n=(-1)^n$ for all $n\ge0$], the requested form of <Cassini's identity>.