Chapter 5.6, Problem 32ES

In 24-34, $F_a,F_1,F_2,\mathrm{...}$is the Fibonacci sequence.

Prove that for each integer $n\ge 0$ , gcd $n\ge 0,\gcd \left(F_{n+1},F_n\right)=1.$ (The definition of gcd is given in Section 4.10)