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# The difference between any two consecutive interior angles of a polygon is 5. If the smallest angle is 120 , find the number of the sides of the polygon.

18.  The difference between any two consecutive interior angles of a polygon is $5 \degree$. If the smallest angle is $120 \degree$  , find the number of the sides of the polygon.

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The angles of polygon forms AP with common difference of $5 \degree$ and first term as $120 \degree$ .

We know that sum of angles of polygon with n sides is $180(n-2)$

$\therefore S_n=180(n-2)$

$\Rightarrow \frac{n}{2}[2a+(n-1)d]=180(n-2)$

$\Rightarrow \frac{n}{2}[2(120)+(n-1)5]=180(n-2)$

$\Rightarrow n[240+5n-5]=360n-720$

$\Rightarrow 235n+5n^2=360n-720$

$\Rightarrow 5n^2-125n+720=0$

$\Rightarrow n^2-25n+144=0$

$\Rightarrow n^2-16n-9n+144=0$

$\Rightarrow n(n-16)-9(n-16)=0$

$\Rightarrow (n-16)(n-9)=0$

$\Rightarrow n=9,16$

Sides of polygon are 9 or 16.

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