Inequalities - AM-GM

Question

Let $H_n = 1 + \frac12 + \frac13 + ... + \frac1n$

Prove that;

$H_n + n$ $\geq$ n$(n+1)^\frac{1}{n}$ for $n$ $\leq$ $2$

I have tried writing $H_n + n = \frac12 + \frac13 +...+ \frac1n + (n+1)$ but am left with an $n!$ in the denominator after applying AM-GM. I have also tried $H_n + n +1 + 2 + 3 + ... +n$ but this does not work either.

I am looking for hints put me on the right track, note, i need to solve this using AM-GM.

Answer 1

HINT

$$\sum_{k=1}^{n}\frac{1}{k}+n=\sum_{k=1}^{n}\frac{k+1}{k}$$

Realize that $\frac{k+1}{k}\times \frac{k+2}{k+1}=\frac{k+2}{k}$.

Now, it's much simpler to use AM-GM since the multiples cancel each other out.