Chapter 12.4, Problem 43PE

For Exercises 43−44, use the Fibonacci sequence $\left\{F_n\right\}=\left\{1,1,2,3,5,8,13,\mathrm{...}\right\}$ . Recall that theFibonacci sequence can be defined recursively as $F_1=1,F_2=1$ , and $F_n=F_{n-1}+F_{n-2}$ for $n\ge 3$ .

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