How do you show $s_n = \frac{x_{n+1}}{x_n}$ where $(x_n)$ is the Fibonacci sequence?

Question

Let $(s_n)$ denote the sequence satisfying: $s_{n+1} = 1 + \frac{1}{s_n}$ with $s_0 = 1$.

Let $(x_n)$ denote the Fibonacci sequence and $x_n = \frac{5 + \sqrt{5}}{10}(\frac{1 + \sqrt{5}}{2})^n + \frac{5 - \sqrt{5}}{10}(\frac{1 - \sqrt{5}}{2})^n$.

How do you show that $s_n = \frac{x_{n+1}}{x_n}$?

when finding a fromula for $s_n$ i reach the equation:

$r^{2n + 1} - r^n - 1 = 0$ and i don't know where to go from there

Answer 1

The base case is clear. Now notice that

$$s_{n+1}=1+1/s_n=\frac{x_{n+1}+x_n}{x_{n+1}}.$$

But $x_{n+1}+x_n=x_{n+2}$, thus the assertion holds by induction.