Continuing fractions: how is this result obtained?

Question

I came across a question that says $$x = [1, \bar{s}]$$ which I understand means $$1+\frac{1}{s+\frac{1}{s+\frac{1}{s+...}}}$$ it then says that $$y = [s, \bar{s}]$$ and to show that y satisfies $$y = s + \frac{1}{y}$$ (which can easily be seen, since y is defined recursively). However it asks to show that $$x = 1+ \frac{2}{s+\sqrt{4+s^2}}$$ based on y satisfying the equation above. I'm really not sure how the final step is reached, let alone have any clue how to prove it, so any pointers would be greatly appreciated.

Answer 1

Notice that $$ x = 1 + \frac{1}{y} \tag{1} $$ Also, from $$ y = s + \frac{1}{y} $$ we get $y^2 - sy - 1 = 0$, so $$ y = \frac{s \pm \sqrt{s^2 + 4}}{2}. $$ $y$ is positive, so it should be in fact $$ y = \frac{s + \sqrt{s^2 + 4}}{2} \tag{2} $$ Now all you have to do is plug (2) into (1). You should get the desired result.

Answer 2

Hint: Solve for $y$ in terms of $s$. Then just observe that $x=y+s-1$ (why?) and…