Answer
See the explanation
Work Step by Step
Let $P(n)$ denote the statement $F_1^2+F_2^2+\ldots+F_n^2=F_nF_{n+1}$.
Step 1.
$P(1)$ is the statement that $F_1^2=F_1F_2$, or $1^2=1\cdot 1$, which is true.
Step 2.
Assume that $P(k)$ is true. Thus our induction hypothesis is $F_1^2+F_2^2+\ldots+F_k^2=F_kF_{k+1}$.
We want to show that $P(k+1)$ is true.
$F_1^2+F_2^2+\ldots+F_{k}^2+F_{k+1}^2=F_kF_{k+1}+F_{k+1}^2$ (By Induction hypothesis)
$=F_{k+1}(F_k+F_{k+1})$ (By Distributive property)
$=F_{k+1}F_{k+2}$ (By the formula $F_n=F_{n-1}+F_{n-2}$ for $n=k+2$)
$=F_{k+1}F_{(k+1)+1}$
Thus $P(k+1)$ follows from $P(k)$, and this completes the induction step.
