The interior angles of an n-sided convex poly | vedclass

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(A) The sum of interior angles of an $n$-sided polygon is given by $(n-2) 	imes 180^\circ$.The angles are in $G.P.$ with first term $a = 1^\circ$ and

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(A) The sum of interior angles of an $n$-sided polygon is given by $(n-2) 	imes 180^\circ$.The angles are in $G.P.$ with first term $a = 1^\circ$ and common ratio $r = 2$.The sum of $n$ terms of a $G.P.$ is $S_n = a \frac{r^n - 1}{r - 1}$.Substituting the values: $S_n = 1 	imes \frac{2^n - 1}{2 - 1} = 2^n - 1$.Equating the two expressions for the sum: $2^n - 1 = (n - 2) 	imes 180$.$2^n - 1 = 180n - 360$.$2^n = 180n - 359$.For any integer $n \ge 3$,the $L.H.S.$ $(2^n)$ is always even,while the $R.H.S.$ $(180n - 359)$ is always odd.Since an even number cannot equal an odd number,there are no possible values for $n$.

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $1^\circ$ and the common ratio is $2$. Then the number of possible values of $n$ is:

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