MathDB

Let f(x) \equal{} \frac{x^2\plus{}1}{2x} for

x \neq 0.

Define f^{(0)}(x) \equal{} x and f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x)) for all positive integers

n

and

x \neq 0.

Prove that for all non-negative integers

n

and x \neq \{\minus{}1,0,1\} \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.

Problem Statement