The interior angles of a convex polygon are in an arithmetic progression. The sum of the three smallest interior angles is 396^ and the sum of the three largest interior angles is 504^ . The number of sides of the polygon is

Step-by-step solution

Let the number of sides of the polygon be n , and let its interior angles in ascending order be A₁, A₂, , A_n . Since the angles are in an arithmetic progression, the sum of terms equidistant from the beginning and the end is constant: A₁ + A_n = A₂ + A_ n-1 = A₃ + A_ n-2 We are given: A₁ + A₂ + A₃ = 396^ A_n + A_ n-1 + A_ n-2 = 504^ Adding these two equations gives: (A₁ + A_n) + (A₂ + A_ n-1 ) + (A₃ + A_ n-2 ) = 396^ + 504^ = 900^ Since each pair sums to the same value, we have: 3(A₁ + A_n) = 900^ A₁ + A_n = 300^