Given $\displaystyle f_c (x)= c - \frac{c^2}{x} $, define $\displaystyle x_{n+1} =f_c (x_n) $. using $\displaystyle x_0 =p$ as an inital value, show that this nonlinear difference equation is a 3 cycle.
It suffices to show $\displaystyle x_3=p$. By expanding out you get $\displaystyle x_3 = c-{c^2\over c-{c^2\over c-{c^2 \over p}}}$, and its fairly straightforward (if tedious) to simplify this to p.