# Math Help - inequality

1. ## inequality

Can anybody help me with the proof of this inequality please?

$sum_{k=1}^n(a_k).sum_{k=1}^n(1/a_k)<=n^2$
$a_k>0$

thank you

2. The correct inequality is this:

$\left(\sum_{k=1}^na_k\right)\left(\sum_{k=1}\frac{ 1}{a_k}\right)\geq n^2$

Use the AM-GM inequality.

$\sum_{k=1}^na_k\geq n\sqrt[n]{a_1a_2\ldots a_n}$

$\sum_{k=1}^n\frac{1}{a_k}\geq n\cdot\frac{1}{\sqrt[n]{a_1a_2\ldots a_n}}$

Now multiply the inequalities, member with member.