The interior angles of a convex polygon are in A.P. If the smallest angle is $100°$ and the common difference is $4°$, then the number of sides is

We know that the sum of the interior angles of a convex polygon of $n$ sides is $(n–2) (180°)$. We are given $a=100$ and $d=4.$
$\begin{array}{l}\therefore \frac{n}{2}\left[2\right(100)+(n-1\left)\right(4\left)\right]=(n-2)\left(180\right)\\ \Rightarrow n^2-41n+180=0\Rightarrow n=5,36\end{array}$
If $n=36,$ then interior angles of the polygon are $100°, 104°, .........,176°, 180°, 184°,......244°.$