• question_answer The interior angles of a polygon are in arithmetic progression. The smallest angle is $\mathbf{12}{{\mathbf{0}}^{{}^\circ }}$ and the common difference is${{\mathbf{5}}^{{}^\circ }}$. What will be the number of sides of the polygon A)   8                   B)  9               C)  10                                D)  7

[b] Let 'a' and "d" be the 1st term & common difference in A.P. respectively. Given, $a={{120}^{{}^\circ }}\And d={{5}^{{}^\circ }}$ $S=\frac{n}{2}[2a+(n-1).d]$ By Geometry, Sum of angles of the polygon be $\left( 2n-4 \right)\times {{90}^{{}^\circ }}$ $\therefore (2n-4)\times {{90}^{{}^\circ }}=\frac{n}{2}[2\times {{120}^{{}^\circ }}+(n-1)5]$ $\Rightarrow {{180}^{{}^\circ }}(2n-4)=240n+5{{n}^{2}}-5n$ 5n2 + 240n - 5n - 360n + 720 = 0 $\Rightarrow 5{{n}^{2}}-125n+720=0\Rightarrow {{n}^{2}}-25n+144=0$$\Rightarrow {{n}^{2}}-16n-9n+144=0\Rightarrow (n-16)(n-9)=0$ $n=9,16$ $\because n=16$              (It is not possible) $\therefore n=9$ Hence, the no. of sides of the polygone, $n=9.$ Hence, option [b] is correct
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