The interior angles of a convex polygon are in an arithmetic progression. If the smallest angle is 120^ and the common difference is 5^ , then the number of sides of the polygon is

Step-by-step solution

Let the number of sides of the polygon be n . The sum of the interior angles of an n -sided polygon is given by (n-2) 180^ . Since the angles are in an arithmetic progression with the first term a = 120^ and common difference d = 5^ , the sum of the n angles is also given by: S_n = n 2 [2a + (n-1)d] Equating the two expressions for the sum: n 2 [2(120) + (n-1)5] = (n-2) 180 n[240 + 5n - 5] = 360(n - 2) 235n + 5n^2 = 360n - 720 5n^2 - 125n + 720 = 0 n^2 - 25n + 144 = 0 (n-9)(n-16) = 0 This gives n = 9 or n = 16 . If