Some Basic induction help.

Question

Let $r\in \Bbb R$ such that $r + \frac{1}{r}\in \Bbb N$, Prove by induction that $r^n + \frac{1}{r^n}\in \Bbb N$ for every $n\in \Bbb N$.

I've done some expansions for $(r+\frac{1}{r})^n$ for $n = 2,3$ and I can see how it holds.

I am having trouble showing it for all $n.$

Answer 1

Hint: Suppose $r^n+\frac{1}{r^n} \in \mathbb{N}$ for some $n \ge 1$. Then $$\left(r^n+\frac{1}{r^n}\right)\left(r+\frac{1}{r}\right) = r^{n+1} + \frac{1}{r^{n+1}} + r^{n-1} + \frac{1}{r^{n-1}}.$$

Answer 2

Hint: What is $(r + \frac{1}{r})^2 $? You can multiple this by $r + \frac{1}{r}$ and get what you want for $n = 3$. Adopt this for general case.