Hi guys, im trying to figure out this problem:

the only step i can figure out is substituting $\displaystyle n$ for $\displaystyle 2^m$, resulting in:Quote:

prove that:

$\displaystyle \frac{1}{n}\displaystyle\sum_{i=1}^n x_i \geq

\left(\displaystyle\prod_{i=1}^n x_i\right)^\frac{1}{n} $

for positive integersnand postive real numbers $\displaystyle x_i$

It does not seem to be possible to give a direct proof of this result using induction on $\displaystyle n$. However, it can be proved for $\displaystyle n = 2^m$ for $\displaystyle m \geq 0 $ by induction on $\displaystyle m $. The general result now follows by proving the converse of the usual inductive step: if the result holds for $\displaystyle n = k +1 $, where $\displaystyle k$ is a positive integer, then it holds for $\displaystyle n = k $.

$\displaystyle \frac{1}{2^m}\displaystyle\sum_{i=1}^{2^m} x_i \geq

\left(\displaystyle\prod_{i=1}^{2^m} x_i\right)^\frac{1}{2^m} $

and then im stuck (after hours of fiddling with this inequality). i hope someone can help, thanks in advance