Question 5
A5 (POL) Let $f(x)=\frac{x^2+1}{2x}$ for $x\ne0$. Define $f^{(0)}(x)=x$ and $f^{(n)}(x)=f(f^{(n-1)}(x))$ for all positive integers $n$ and $x\ne0$. Prove that for all nonnegative integers $n$ and $x\ne-1,0,$ or $1$,
$$\frac{f^{(n)}(x)}{f^{(n+1)}(x)}=1+\frac1{f\left(\left(\frac{x+1}{x-1}\right)^{2^n}\right)}.$$
$$\frac{f^{(n)}(x)}{f^{(n+1)}(x)}=1+\frac1{f\left(\left(\frac{x+1}{x-1}\right)^{2^n}\right)}.$$