Chapter 8.4, Problem 47PE

For Exercises 43-44, use the Fibonacci sequence $\left\{F_n\right\}=\left\{1,1,2,3,5,8,13,\dots \right\}$

. Recall that the Fibonacci sequence can be defined recursively as $F_1=1,F_2=1$, and $F_n=F_{n-1}+F_{n-2}$

for $n\ge 3$.

Prove that $F_1+F_2+F_3+\dots +F_n=F_{n+2}-1$ for positive integers $n\ge 3$.