The intrerior angles of a polygon are 8,12,18,...... Calculate the sides of polygon

Question

The angles are in the ascending order

Answer 1

8,12,18,... may represent a geometric progression (GP) where a=8 and r=1.5 and the series up to the nth term, where integer n>2, is:

a,ar,ar²,ar³,...,ar^n-1. The sum S_n=a(rⁿ-1)/(r-1).

The sum of the interior angles of a polygon (n-gon) is 180(n-2)°.

Therefore if the angles form a geometric progression then:

S_n=a(rⁿ-1)/(r-1)=180(n-2) for some integer n.

For a=8 and r=1.5:

8(1.5ⁿ-1)/0.5=16(1.5ⁿ-1)=180(n-2),

1.5ⁿ=11.25(n-2)+1=11.25n-21.5. This has no integer solutions for n.

So the given set of terms cannot be a GP. Neither do the terms form an arithmetic progression (AP).

Even if the series was 6,12,18,... (AP), n could not be an integer. 

More information is needed.