Answer
See the explanation
Work Step by Step
Let $P(n)$ denote the statement $F_1+F_2+\ldots+F_n=F_{n+2}-1$.
Step 1.
P(1) is the statement that $F_1=F_3-1$, or $1=2-1$, which is true.
Step 2.
Assume that $P(k)$ is true. Thus our induction hypothesis is $F_1+F_2+\ldots+F_k=F_{k+2}-1$.
We want to show that $P(k+1)$ is true.
That is,
$F_1+F_2+\ldots+F_k+F_{k+1}=(F_{k+2}-1)+F_{k+1}$ (By Induction hypothesis)
$=(F_{k+1}+F_{k+2})-1$ (Use the formula $F_{n}=F_{n-1}+F_{n-2}$ for $n=k+3$)
$=F_{k+3}-1$
$=F_{(k+1)+2}-1$
Thus, $P(k+1)$ follows from $P(k)$, and this completes the induction step.
