Question 97
Let $x$ and $y$ be real numbers, and define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ by
$$
a_{n}=\sum_{k=0}^{n} x^{k} y^{n-k}
$$
Suppose that $a_{m}, a_{m+1}, a_{m+2}, a_{m+3}$ are integers for some $m \geq 0$. Prove that $a_{n}$ is an integer for all $n \geq 0$.
$$
a_{n}=\sum_{k=0}^{n} x^{k} y^{n-k}
$$
Suppose that $a_{m}, a_{m+1}, a_{m+2}, a_{m+3}$ are integers for some $m \geq 0$. Prove that $a_{n}$ is an integer for all $n \geq 0$.