# Math Help - inequality relating to arithmetic and geometric means

1. ## inequality relating to arithmetic and geometric means

Hi I am having trouble proving the following inequality:
Let n be a natural number.
$n^{\frac{1}{n}} + n^{\frac{1}{(n+1)}} + ... + n^{\frac{1}{(2n - 1)}} \geq n^{\sqrt[n]{2}}$
The book I found this problem in suggests the use of these mean values:
$A(x_1,x_2,...,x_n) = \frac{x_1+x_2+...+x_n}{n}$
and
$G(x_1,x_2,...,x_n) = \sqrt[n]{x_1x_2...x_n}$
I have tried everything but cant get it to work!
Any suggestions?

2. A previous section before this problem has this fact:
If u,v are positive real numbers and n1, n2 are fixed natural numbers, then
$\sqrt[n1 + n2]{u^{n1}v^{n2}} \leq \frac{n1(u) + n2(v)}{n1 + n2}$
and thus for any two real numbers a and b, where a + b = 1, we have $u^av^b\leq au +bv$

I tried using these facts as well but have still not gotten anywhere.