Answer
$P(n)$ is true for all values of $n$.
Work Step by Step
We need to prove that $F_1+F_2+F_3+......F_n=F_{n+2}-1$
1. Our aim is to find that $P(n)$ is true for $n=1$
$1=F_{1+2}-1 \implies 1 =1$
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 $n=k+1$
$F_1+F_2+F_3+......F_k=F_{k+2}-1 \implies F_{k+2}+F_{k+1}-1=F_{k+3} -1$
This yields: $ F_{k+3}-1=F_{k+3}-1$
So, it is true for $n=k+1$.
Therefore, $P(n)$ is true for all values of $n$.
