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 $n.$

That is wrong for $r=1/2$, since $r+1/r=5/2\not\in\mathbb N$. Also, please tell us what you tried so far once you got the correct problem statement. — martin.koeberl, Mar 27 '17 at 17:14
Is this right? If you let $r=0.1$, then $0.1 + \frac{1}{0.1} = 10.1$. — Ralff, Mar 27 '17 at 17:15
Related : http://math.stackexchange.com/questions/936479/proving-that-frac-phi4001-phi200-is-an-integer — lab bhattacharjee, Mar 27 '17 at 17:48

score 2 · Accepted Answer · answered Mar 27 '17 at 17:17

2

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

answered Mar 27 '17 at 17:17

angryavian

89,882

score 1 · Answer 2 · answered Mar 27 '17 at 17:16

1

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.

answered Mar 27 '17 at 17:16

Yunus Syed

1,737

Some Basic induction help.

2 Answers2