- #1
threeder
- 27
- 0
Homework Statement
Prove that
[tex]\frac{1}{n}\sum_{i=1}^n x_i\geq {(\prod_{i=1}^n x_i)}^{1/n}[/tex]
for positive integers [itex]n[/itex] and positive real numbers [itex]x_i[/itex]
Homework Equations
There is also a hint. It states that it does not seem to be possible to prove it directly so you should prove it for [itex]n=2^m, m\geq 0[/itex]. Nonetheless, the result should follow by proving the converse of the usual inductive step: if it hold for [itex]n=k+1[/itex], the it also does for [itex]n=k[/itex]
The Attempt at a Solution
I don't really understand how should I proceed. I rewrote the inequality
[tex]\frac{1}{2^{k+1}}\sum_{i=0}^{2^{k+1}} x_i\geq {(\prod_{i=0}^{2^{k+1}} x_i)}^{2^{-k-1}}[/tex]
but that's it. Can't seem to understand how can we proceed from the converse using the induction principe. Any hints would be apreciated. Thanks!
Last edited: