# 1968 AHSME Problems/Problem 20

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## Problem

The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals:

$\text{(A) } 9\quad \text{(B) } 10\quad \text{(C) } 12\quad \text{(D) } 16\quad \text{(E) } 32$

## Solution

The formula for the sum of the angles in any polygon is $180(n-2)$. Because this particular polygon is convex and has its angles in an arithmetic sequence with its largest angle being $160$, we can find the sum of the angles.

$a_{n}=160$

$a_{1}=160-5(n-1)$

Plugging this into the formula for finding the sum of an arithmetic sequence...

$n(\frac{160+160-5(n-1)}{2})=180(n-2)$.

Simplifying, we get $n^2+7n-144$.

Since we want the positive solution to the quadratic, we can easily factor and find the answer is $n=\boxed{9}$.

Hence the answer is$\fbox{A}$

## See also

 1968 AHSME (Problems • Answer Key • Resources) Preceded byProblem 19 Followed byProblem 21 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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