Prove that if the sequence $a_0,a_1,a_2,\dots,a_n$ of positive real numbers is log-concave, then it is unimodal I need some hints starting this problem.  It's not that I don't mathematics it's just that I'm not really into the "proof stuff".  I think advanced mathematics is not really my thing...
After I had my first combinatorics class I feel like I'm an "idiot" and I didn't ever think that math can be that hard.
But, anyway, here's the question
Prove that if the sequence $a_0,a_1,a_2,\dots,a_n$ of positive real numbers is log-concave, then it is unimodal.
Thanks!
 A: Note that you mentioned that the sequence is log-concave, therefore:
$$\frac{a_1}{a_0}\geq\frac{a_2}{a_1}\geq\frac{a_3}{a_2}\geq\cdots\geq\frac{a_{n-1}}{a_n}\text{,}$$
which means that ratio $\frac{a_n}{a_{n-1}}$ is steadily decreasing, so it's going to stick below $1$ when it goes below $1$  Therefore, when the sequence of $a_n$ start to decrease, it will just keep decreasing, thus showing that this is unimodal.
$\square$
A: Because it's log-concave, $a_i^2\ge a_{i-1}a_{i+1}$ for all $i$.  We can rearrange this to $\frac{a_{i+1}}{a_i}\le \frac{a_i}{a_{i-1}}$.  This ratio determines if a term is bigger than the next or smaller.  If this ratio starts smaller than 1, then it remains smaller than 1, so the sequence is decreasing.  If this ratio starts bigger than 1, and stays bigger than 1, then the sequence is increasing.  If this ratio starts bigger than 1 and eventually shrinks to smaller than 1, it then stays smaller than 1.  Then, the sequence increases for a while, then decreases from that point on.
