Proof regarding the golden succession

Question

Considering the golden number $\varphi=\frac{1+\sqrt 5}2$ and the succession defined by

$$r_n=1 +\frac 1{r_{n-1}}$$ for all $n \geq 2$ where $r_1=1$

How do I prove that $r_n=\frac{f_n}{f_{n-1}}$, where $f_n$ is the Fibonnaci sequence?

Answer 1

Let $s_n = \frac {f_n}{f_{n-1}}$

We have $$s_n = \frac {f_{n-1}+f_{n-2}}{f_{n-1}}$$

$$= 1+\frac {f_{n-2}}{f_{n-1}} = 1 +\frac {1}{s_{n-1}}$$

Since $s_1= 1=r_1$, we have $s_n = r_n$