## n - gon With Sides in Geometric Progressions

How many polygons may have sides in a geometric progression?
Suppose we have a triangle. The sides are
$a, \: ar, \: ar^2$
. The longest side must be less than the sum of the two shorter sides, so

$a \lt ar+ar^2$
,
$r \lt 1$

$a + ar \lt ar^2$
,
$r \lt 1$

Cancelling
$a$
and solving both inequalities gives
$\frac{\sqrt{2}-1}{2} \lt r \lt \frac{\sqrt{2}+1}{2}$

We can perform the same analysis for each n - gon. Suppose the n gon has an infinite number of sides. Then
$a \lt ar+ar^2+ ...+ ar^{n-1}+...$
,
$r \lt 1$

$a + ar+...+ar^{n-1}+... \lt ar^2$
,
$r \lt 1$

Cacelling
$a$
as before and solving gives
$\frac{1}{2} \lt r \lt 2$
.