The interior angles of a polygon with n sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then n is equal to _____.

$\begin{array}{l}\frac{\mathrm{n}}{2}(2\mathrm{a}+(\mathrm{n}-1\left)6\right)=(\mathrm{n}-2)\cdot {180}^{\circ }\\ \text{ an }+3{\mathrm{n}}^2-3\mathrm{n}=(\mathrm{n}-2)\cdot {180}^{\circ }............\left(1\right)\end{array}$

Now according to question

$\begin{array}{l}\mathrm{a}+(\mathrm{n}-1)6^{\circ }={219}^{\circ }\\ \Rightarrow \mathrm{a}={225}^{\circ }-6{\mathrm{n}}^{\circ }...........\left(2\right)\end{array}$

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

$\begin{array}{l}\left(225\mathrm{n}-6{\mathrm{n}}^2\right)+3{\mathrm{n}}^2-3\mathrm{n}=180\mathrm{n}-360\\ \iff 2{\mathrm{n}}^2-42\mathrm{n}-360=0\\ \Rightarrow \mathrm{n}2-14\mathrm{n}-120=0\\ \mathrm{n}=20,-6\text{ (rejected) }\end{array}$