Check if a system admits solutions of period 2

Question

I have the following problem.

Let $r \geq 0$ be a parameter in the discrete time system $x(k + 1) = r − rx(k)$. Verify whether there exist $r \geq 0$ such that this system admits solutions of period 2.

I do not quiet understand how I should solve this problem. I already determined that the fixed points of the system are $x^* = \frac{r}{r + 1}$ and that $0 \leq r < 1$ for the system to converge.

Answer 1

You should write $x(k+2)=r-rx(k+1)$, substitute in your expression for $x(k+1)$ and see if there are solutions with $x(k+2)=x(k)$