The interior angles of a convex polygon are in arithmetic progression. The smallest angle is 120 degrees and the common difference is 5 degrees. Find the number of sides of the polygon.
+129852 Sum of this arithmetic series = n ( a1 + an) / 2
n = number of sides
a1= 120 = first term
an = 120 + 5 (n -1) = 115 + 5n = last term
So we have n [ 235 + 5n ] / 2 (1)
And the sum of the interior angles of an n-sided polygon = (n - 2) (180) (2)
Set (1) = (2)
n [ 235 + 5n ] /2 = (n -2) (180) simplify
235n + 5n^2 = 2 ( n -2)(180)
235n + 5n^2 = (n -2) (360)
235n + 5n^2 = 360n - 720
5n^2 - 125n + 720 = 0 divide through by 5
n^2 - 25n + 144 = 0 factor
(n - 16) ( n - 9) = 0
Setting each factor to 0 and solve for n and we get two answers
n = 9 or n = 16
