Question

Angles of a polygon are in ap with common difference 5 degree with the smallest angle is 120 degree find the number of sides of polygon

Answer 1

Sum of the interior angles of any polygon is given by 180(n-2) ; where n is the number of sides.
It is given that the interior angles of the polygon are in A.P. and sum of an A.P. is given by
$\frac{n}{2} (2a + (n - 1)d)$
in here
$a = 120 \\ d = 5$
Both the formulae gives the sum of interior angles.
$\frac{n}{2} (240 + (n - 1)5) = 180(n - 2)$
solving the above equation we get a quadratic
${n}^{2} - 25n - 144 = 0$
by applying quadratic formula we get the value of n to be 9 and 16.
Therefore the condition in the question gives two polygons whose number of sides are 9 and 16