Problem

Source: 2009 Romania JBMO TST 3.2

Tags: number theory, Integer, finite set



Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.