Question 151
Let $P$ be a polynomial with rational coefficients of degree greater than or equal to $2$, and let $\left(q_{n}\right)_{n \in \mathbb{N}}$ be a sequence of rationals such that for all $n \geqslant 0$, $q_{n} = P\left(q_{n+1}\right)$. Show that the sequence $q_{n}$ is periodic from some index onward.