The interior angles of a polygon are in A.P. If the smallest angle is $100^{o}$ and the common difference is $4^{o}$ , find the number of sides ?

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Solution

$s u m o f i n t e r i o r a n g l e s o f a n n s i d e d p o l y g o n = 180 (n - 2)$
Angles are in $A P$

Smallest angle $= a = 100^{0}$

Common difference $= d = 4^{0}$

Sum of $n$ terms $\frac{n}{2} [2 a + (n - 1) d]$

Sum of $n$ terms $\frac{n}{2} [200 + (n - 1) 4]$

$s u m o f t e r m s \Rightarrow n [100 + 2 (n - 1)]$ ......(2)

$(1) = (2)$

$\Rightarrow n [100 + 2 (n - 1)] = 180 (n - 2)$

$\Rightarrow 100 n + 2 n^{2} - 2 n = 180 n - 360$

$\Rightarrow 2 n^{2} = 82 n - 360$

$\Rightarrow n^{2} - 41 n + 180 = 0$

$\Rightarrow n^{2} - 36 n - 5 n + 180 = 0$

$\Rightarrow n (n - 36) - 6 (n - 36) = 0$

$\Rightarrow (n - 5) (n - 36) = 0$

$\Rightarrow n = 5$ or $36$

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