# The interior angles of a polygon are in A.P., the smallest angle is 75° and the common difference is 10°. Find the number of sides of the polygon.

Given: The smallest angle is 75°

i.e. a = 75

and common difference = 10°

i.e. d = 10

Therefore, the series is

75, 85, 95, 105, …

and the sum of interior angles of a polygon =(n – 2) 180°

i.e. Sn = 180

We know that,

(n – 2)360 = n [140+10n]

360n – 720 = 140n + 10n2

36n – 72 – 14n – n2 = 0

n2 – 22n + 72 = 0

n2 – 18n – 4n + 72 = 0

n(n – 18) – 4(n – 18) = 0

(n – 4)(n – 18) = 0

Putting both the factor equal to 0, we get

n – 4 = 0 or n – 18 = 0

n = 4 or n = 18

Hence, the number of sides of a polygon can be 4 or 18.

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