# Math Help - Induction

1. ## Induction

Prove that $\frac{1}{n} \sum_{i=1}^{n} x_i \geq \left(\prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}$ for positive integers $n$ and positive real numbers $x_i$.

I dont think I can do this directly with induction on $n$. I let $n = 2^{m}$ for $m \geq 0$. So the induction hypothesis is the following: $\frac{1}{2^{m}} \sum_{i=1}^{2^{m}} x_i \geq \left(\prod_{i=1}^{2^{m}} x_i \right)^{\frac{1}{2^{m}}}$. In other words, I have to prove that $P(k+1) \Rightarrow P(k)$. I would use strong induction then?

Any help is appreciated. Thanks.