interior angles of a convex polygon are in arithematic progression. least angle is 140.find number of sides?

Question

find number of sides of the polygon.

Answer 1

The sum of the interior angles of a convex n-gon is 180 (n - 2)

For example:

n=3 (triangle) 180
n=4 (quadrilateral) 360

So 140 + (140 + d) + ... + (140 + (n-1)d) = 180(n - 2)
140n + (1/2) d n (n-1) = 180n - 360
360 + (1/2) d n (n-1) = 40n

9 + (1/80) d n (n-1) = n
d = 80 (n - 9) / (n (n-1))
d >= 0

n=9 works if the arithmetic progression has difference d=0 (i.e., a 9-gon with all angles equal to 140 degrees).

n=10 works if d = 8/9

n=11 works if d = 16/11

etc.