Question 46
Let $x_0, x_1, \dots, x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 \gt d_2 \gt \dots \gt d_k$ and $\gcd(d_1, d_2, \dots, d_k) = 1$. For every integer $n \ge n_0$, define
$$
x_n = \left\lfloor \frac{x_{n-d_1} + x_{n-d_2} + \dots + x_{n-d_k}}{k} \right\rfloor.
$$
Show that the sequence $\{x_n\}$ is eventually constant.
$$
x_n = \left\lfloor \frac{x_{n-d_1} + x_{n-d_2} + \dots + x_{n-d_k}}{k} \right\rfloor.
$$
Show that the sequence $\{x_n\}$ is eventually constant.