$\text{ The interior angles of a convex polygon forms on A.P with common }$$\text{ difference of }{4}^{\circ }\text{. If the largest interior angle is }{172}^{\circ }\text{ then the number of sides is }$

Let n- be the number of sides of a polygon

$\begin{array}{l}{\mathrm{t}}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1)\mathrm{d}=172\\ \to \mathrm{a}+4(\mathrm{n}-1)=172\\ \to \mathrm{a}=172-4(\mathrm{n}-1)-\cdots -\left(1\right)\\ {\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}[\mathrm{a}+172]=(\mathrm{n}-2)\left(180\right)---\left(2\right) \left(\sin ce sum of the interior angles of a \text{'}n\text{'}sided polygon=\left(n-2\right){180}^{°}\right)\\ \text{ From}\left(1\right)\text{ and}\left(2\right)\Rightarrow \mathrm{n}=12.\end{array}$