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Question

The measures of the interior angles taken in order of a polygon form an arithmetic sequence. The least measurement in the sequence is 85. The greatest measurement is 215. Find the number of sides in the given polygon

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Solution

Let n denote the number of sides of the polygon.
Now, the measures of interior angles form an arithmetic sequence.
Let the sum of the interior angles of the polygon be
Sn=a+(a+d)+(a+2d)+....+l, where a=85 and l=215.
We have, Sn=n2[l+a]...(1)
We know that the sum of the interior angles of a polygon is (n2)×180.
Thus, Sn=(n2)×180
From (1), we have n2[l+a]=(n2)×180
n2[215+85]=(n2)×180
150n=180(n2)n=12.
Hence, the number of sides of the polygon is 12.

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