Question

The interior angle of a polygon are in ap .The smallest angle is 52 deger and the common difference is 8 deger find the number of sides of the polygon

Answer 1

52+(52+8)+(52+8+8)+(52+8+8+8)
52,60,68,76

Answer 2

Given smallest interior angle is 52°Common difference is 8° Each of the interior angle increases by 8° Hence exterior decreases by 8° Exterior angle corresponding to 52° = 180° - 52° = 128°Next exterior angle = 128° - 8° = 120° and so on Hence the AP is 128°, 120°… Sum of exterior angles of a polygon = 360° That is 128° + 120° +… n terms = 360° Sum of n terms of AP is (n/2)[2a + (n – 1)d] Hence, 360° = (n/2)[2(128°) + (n – 1)(-8°)] 720 = n(256 – 8n + 8) 8n2 – 264n + 720 = 0 n2 – 33n + 90 = 0 n2 – 30n – 3n + 90 = 0 n(n – 30) – 3(n + 30) = 0 (n – 30)(n – 3)=0 n – 30 = 0 or n – 3 = 0 Thus n = 30 or n = 3But n cannot be 30 as the exterior angles become negative after few terms. Thus n = 3