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.