Maths Questions from AP (Arithmetic Progression)

maths

#1

The interior angles of a polygon are in AP. The smallest angle is 520 and the common difference is 80. Find the number of sides of the polygon.


#2

Are you sure this is the correct question? The smallest angle cannot be 520, because even if it is a concave polygon, there has to be angles that are acute angles. I think your question is

The interior angles of a polygon are in AP. The smallest angle is 52 degrees and the common difference is 8 degrees. Find the number of sides of the polygon.

Assume there are n sides and solve for n
Sum = n/2 [ 2a+(n-1) d]

Since the interior angles are in arithmetic progression with first term a=52 and common difference d=8,

180(n-2) = n/2 [ 2(52)+(n-1) 8]
Multiply both sides by 2
360(n-2) = n[ 104 +8n-8]
360n-720 = 104n +8n^2-8n

Rearrange the equation as a quadratic.

8n^2+96n-360n-720=0
8n^2-264n-720=0
n^2-33n-90=0
n^2-30n-3n-90=0
n(n-30)-3(n-30)=0
(n-30)(n-3) = 0
n=30 or n=3