The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is

Question

Smart Achievers · Accepted Answer

(1)

$Let angles of a polygon are a, a + d, a + 2 d, \dots$

$a = 120 °,$ $d = 5$

$So series are 120 °, 125 °, 130 °, \dots$

$Sum of A.P. = (n - 2) 180 °$

$\Rightarrow$ $\frac{n}{2} [2 a + (n - 1) d] = (n - 2) 180 °$

$\Rightarrow$ $\frac{n}{2} [240 + (n - 1) 5] = 180 (n - 2)$

$\Rightarrow$ $5 n^{2} - 125 n + 720 = 0 \Rightarrow n^{2} - 25 n + 144 = 0$

$\Rightarrow$ $n = 16,$ $n = 9 not possible.$

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