maths 10分

一多邊形的內角成一公差為 5° 之等差數列 , 已知最大角是160° 。 a)Let n be the number of sides of the polygon :Consider the sum of the interior angles :
(n - 2)(180°) = 160° + (160° - 5°) + (160° - 5(2)°) + ... + (160° - 5(n-1)°)
(n - 2)(180) = (n/2) ( 2*160 - 5(n-1) )
(n - 2)(36) = (n/2) ( 64 - (n-1) )
36n - 72 = (n/2) (65 - n)
72n - 144 = 65n - n²
n² + 7n - 144 = 0
(n - 9)(n + 16) = 0
n = 9 or n = - 16 (rejected)The number of sides of the polygon is 9.
b)The smallest interior angle = 160° - 5(n-1)°
= 160° - 5(9 -1)°
= 120°