Let x be a real number and have the continued fraction expansion $x = [1,\bar{s}]$, where s is a natural number and $\bar{s}$ denotes the infinite string $s, s, \ldots $
Let $y = [s, \bar{s}]$ and show $y$ satisfies
$\displaystyle y = s + \frac{1}{y}$
Hence, show that,
$\displaystyle x = 1+ \frac{2}{s+\sqrt{4+s^2}}$
For the first part, I wrote $y =s + \cfrac{1}{s + \cfrac{1}{s + \cfrac{1}{s + \ddots}}} $, then took $\frac{1}{y-s} = \cfrac{1}{\cfrac{1}{s + \cfrac{1}{s + \ddots}}}$, which returned the expansion for y,
meaning y = $\frac{1}{y-s}$
, y = s + $\frac{1}{y}$.
The second part implies that it uses the first, so I tried writing
$\displaystyle x = 1 + \cfrac{1}{s + \cfrac{1}{s + \ddots}}$
and substituting for $y$ to get $x = 1 + (y-s)$, but I could not get further.