Chapter 12.1, Problem 53E

To determine

To calculate: Thenumber of sides of the polygon.

Answer to Problem 53E

The number of sides of the polygonare $12$ .

Explanation of Solution

Given information:

The angles of a convex polygon form an arithmetic sequence such that least angle in sequence is $128°$ and greatest angle in sequence is $172°$ .

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the provided situation that angles of a convex polygon form an arithmetic sequence such that least angle in sequence is $128°$ and greatest angle in sequence is $172°$ .

Denote the least angle as first term of arithmetic sequence $a_1$ and greatest angle as last term $a_n$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

According to question $a_1=128$ and $a_n=172$ .

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ 172=128+\left(n-1\right)d\\ \left(n-1\right)d=44\end{array}$

Sum of angles of a convex polygon with n sides is $\left(n-2\right)\cdot 180$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Apply it and use $\left(n-1\right)d=44$ .

  $\begin{array}{c}\left(n-2\right)\cdot 180=\frac{n}{2}\left[2\left(128\right)+44\right]\\ 180n-360=\frac{n}{2}\left[300\right]\\ 180n-360=150n\\ 30n=360\end{array}$

Simplify it further as,

  $n=12$

Thus, the number of sides of the polygonare $12$ .