The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then n is equal to

Question

Aicharya · Accepted Answer

$\frac{n}{2}[2 a+(n-1) 6]=(n-2) 180^{\circ}$
and $a n+3 n^2-3 n=3 n(n-2) 180^{\circ}$...(i)
$\begin{aligned}
& \therefore \quad \text { Given } a+(n-1) 6^{\circ}=219^{\circ} \\ & \Rightarrow a=225^{\circ}-6 n^{\circ}
\end{aligned}$
Putting value of $a$ in (i)
$\begin{aligned}
& \text { We get }\left(225-6 n^2\right)+3 n^2-3 n=180 n-360^{\circ} \\ & \Rightarrow 2 n^2-42 n-360=0 \\ & \Rightarrow n^2-14 n-120=0 \\ & \Rightarrow(n-20)(n+6)=0 \\ & \Rightarrow n=20,-6 \text { (Rejected) } \\ & \therefore n=20
\end{aligned}$

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}$. If the…

Solution