Question

# The measures of the angles of a convex polygon form an arithmetic sequence. The least measurement in the sequence is 129°. The greatest measurement is

Sequences
The measures of the angles of a convex polygon form an arithmetic sequence. The least measurement in the sequence is 129&deg;. The greatest measurement is 159&deg;. Find the number of sides in this polygon.

2021-05-17
We are given a1=129 (least angle) and an=159 (greatest angle, representing the nnth angle). Since the number of angles is equal to the number of sides in a convex polygon, we need to find n.
Use the formula for finding the sum of the first nn term of an arithmetic sequence:
$$\displaystyle{S}{n}=\frac{{n}}{{2}}{\left({a}{1}+{a}{2}\right)}$$
$$\displaystyle{S}{n}=\frac{{n}}{{2}}{\left({129}+{159}\right)}$$
PSKSn=144n
The sum of the angles of a convex polygon with nn sides (in degrees) is given by: Sn=180(n−2)
Equating (1) and (2), we solve for nn: 144n=180(n−2)
144n=180n−360
−36n=−360
n=10
So, the convex polygon has 10 sides.