Let
![h_n](/media/m/4/2/e/42ede44fa1f7ee4745f91f4deeb11236.png)
be the apothem (distance from the center to one of the sides) of a regular
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
-gon (
![n \geq 3](/media/m/5/4/8/54807b3bf99aa939833fe57bf8d891d3.png)
) inscribed in a circle of radius
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
. Prove the inequality
![(n + 1)h_n+1 - nh_n > r.](/media/m/9/f/5/9f528d5e8ff2a10845c5c4870d46ebdf.png)
Also prove that if
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
on the right side is replaced with a greater number, the inequality will not remain true for all
%V0
Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality
$$(n + 1)h_n+1 - nh_n > r.$$
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$