Solution
= Solution
Use <mathematical induction> with the defining <Fibonacci recurrence>. Both identities hold at $n=0$: their two sides are zero. If the first holds at $n$, then
$$
\sum_{i=0}^{n+1}F_i
=F_{n+2}-1+F_{n+1}=F_{n+3}-1.
$$
If the second holds at $n$, then
$$
\sum_{i=0}^{n+1}F_i^2
=F_nF_{n+1}+F_{n+1}^2
=F_{n+1}(F_n+F_{n+1})
=F_{n+1}F_{n+2}.
$$
Each <inductive hypothesis> therefore advances to $n+1$.