The interior angles of a convex polygon are in an arithmetic progression. If the smallest angle is $100^{\circ}$ and common difference is $4^{\circ}$, then find the number of sides.
+128408 Let the first angle = 100
And let the greatest angle be ( 100 + 4(n -1) ) where n is the number of sides
Sum of the interior angles = 180 ( n - 2)
So....using the formula for the sum of an arithmetic progression we have that
(100 + 100 + 4 ( n - 1) ) ( n /2) = 180 ( n - 2)
( 200 + 4n - 4) ( n/2) =180 (n - 2)
(196 + 4n) (n /2) =180 ( n- 2)
(98 + 2n)n = 180 n - 360
98n + 2n^2 = 180n - 360
2n^2 - 82n + 360 = 0
n^2 - 41n + 180 = 0
(n - 45) ( n + 4) = 0
Setting the first factor to 0 and solving for n gives
n = 45 = the number of sides
