Why $\frac{F_{n+2}}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}$?

Question

Why $\frac{F_{n+2}}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}$ (seems to) hold for every fibonacci number $F_n$?

Answer 1

We know from the definition of the Fibonacci sequence that $F_{n+2}=F_{n+1}+F_{n}$.

Suppose that $n$ is such that $F_{n+1}\neq 0$.

Dividing both sides of the recursive formula given in the definition by $F_{n+1}$ yields the result.