Solution
= Solution
We prove both identities by <mathematical induction>. They hold at $n=1$. Assume they hold at $n$. Using $F_{n-1}=F_{n+1}-F_n$,
$$
\begin{aligned}
F_{2n+2}&=F_{2n+1}+F_{2n}\\
&=F_n^2+F_{n+1}^2+F_n(F_{n-1}+F_{n+1})\\
&=F_{n+1}(F_n+F_{n+2}).
\end{aligned}
$$
Adding this to $F_{2n+1}=F_n^2+F_{n+1}^2$ and using $F_{n+2}=F_n+F_{n+1}$ gives
$$
F_{2n+3}=F_{n+1}^2+F_{n+2}^2.
$$
These are precisely the two assertions with $n$ replaced by $n+1$, completing the induction.