Let $r_1$, $\ldots$, $r_k$ distinct (non-zero) rational numbers, $P_1$, $\ldots$, $P_k \in \mathbb{Q}[x]$ non-zero polynomials, such that for every integer $n\ge 0$ the number
$$\sum_{s=1}^k P_s(n) r_s^n$$
is an integer.
Show that all of the numbers $r_1$, $\ldots$, $r_k$ are integers.
$\bf{Notes:}$ This is a particular case of a question I posted before. The solution I managed to come up with of the general case could be understood here, even by people not familiar with the rudiments of algebraic number theory. Thus the posting. I would like to see whether there are any other distinct solutions in this case. Thank you for your attention! Any feedback would be appreciated!
