Let $r$ be a real number such that $r + 1/r$ is an integer. Prove that for every natural number $n$, $r^n + 1/r^n$ is also an integer. (In addition, I have to use induction, strong induction, or a minimum counterexample).
I initially tried minimum counterexample, assuming that for a fixed $r$, that $k$ is the smallest natural number for which the statement is false. I then had: $$r^{k-1} + (1/r)^{k-1} = n \in \mathbb{N}$$ with the intent of showing a contradiction for the case $k$, but couldn't get anywhere after combining the terms into a single fraction. Similar attempts with induction and strong induction went nowhere.
All and any help is appreciated. Thank you kindly!
First note that that the result is true for $k=0$. It is also true for $k=1$ (by hypothesis)
For $k \geq 2$ \begin{eqnarray*} r^{k}+\frac{1}{r^{k}}=(r+\frac{1}{r})(r^{k-1}+\frac{1}{r^{k-1}})-(r^{k-2}+\frac{1}{r^{k-2}}) \end{eqnarray*} & so the result follows by induction.