Answer
$P(n)$ is true for all values of $n$.
Work Step by Step
We need to prove that $F_1^2+F_2^2+F_3^2+......F_n^2=F_{n+1}F_n$
1. Our aim is to find that $P(n)$ is true for $n=1$
$F_1^2=F_{1+1}F_1 \implies 1 =1$
So, it is true for $n=1$.
2. Our aim is to find that $P(n)$ is true for $n=k$.This, it will also true for $n=k+1$
$(F_1^2+F_2^2+F_3^2+......F_k^2)+F_{k+1}^2=F_{(k+1)+1}F_{k+1} \implies (F_{k+1})(F_k+F_{k+1})=F_{k+2} F_{k+1}$
This yields: $ (F_{k+1})(F_{k+2})=F_{k+1} F_{k+2}$
So, it is true for $n=k+1$.
Therefore, $P(n)$ is true for all values of $n$.
