Problem: The measures of the interior angles of a convex polygon of n sides are in arithmetic progression. If the common difference is 5∘ and the largest angle is 160∘, then n equals:
Answer Choices:
A. 9
B. 10
C. 12
D. 16
E. 32
Solution:
Method I. a+(n−1)5=160,a=160−(n−1)5
(n−2)180=21n[2a+(n−1)5]=21n[320−(n−1)5]
n2+7n−144=0=(n+16)(n−9)∴n=9.
Method II. The exterior angles are 20,25,30,…; their sum is 360∘
∴360=21n[40+(n−1)5]∴n2+7n−144=0∴n=9.