The interior angles of a polygon with n sides, are in an A.P.

Question

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 \_\_\_\_\_\_\_.

Answer: 20

Accepted Answer

$\begin{gathered} \frac{n}{2}(2a+(n-1)6)=(n-2)\cdot {180}^{\circ } \ an+3{n}^{2}-3n=(n-2)\cdot {180}^{\circ }\, \end{gathered}$

Now according to question

$\begin{gathered} a+(n-1){6}^{\circ }={219}^{\circ } \ \Rightarrow a={225}^{\circ }-6{n}^{\circ }\, \end{gathered}$

Putting value of a from equation (2) in (1)

We get

$\begin{gathered} (225n-6{n}^{2})+3{n}^{2}-3n=180n-360 \ \Rightarrow 2{n}^{2}-42n-360=0 \ \Rightarrow n2-14n-120=0 \ n=20,-6 \end{gathered}$

Solutions · 1