Question 84
Let $\alpha$ and $\beta$ be distinct real numbers. Consider the sequence $A_1, A_2, \dots$ of real numbers given by
$$
A_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}
$$
for $n \ge 1$. If the members $A_p, A_{p+1}, A_{p+2}$ are integers for some prime number $p$, prove that all members of the sequence are integers.
(Proposed by Bayarmagnai Gombodorj)
$$
A_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}
$$
for $n \ge 1$. If the members $A_p, A_{p+1}, A_{p+2}$ are integers for some prime number $p$, prove that all members of the sequence are integers.
(Proposed by Bayarmagnai Gombodorj)