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 $(n-2) 	imes 180^\circ$.Since the angles are in $G.P.$ with first term $a = 120^\c

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(A) The sum of interior angles of an $n$-sided polygon is given by $(n-2) 	imes 180^\circ$.Since the angles are in $G.P.$ with first term $a = 120^\circ$ and common ratio $r = 2$,the sum of $n$ terms is $S_n = a(r^n - 1) / (r - 1)$.Substituting the values: $S_n = 120(2^n - 1) / (2 - 1) = 120(2^n - 1)$.Equating the two expressions: $120(2^n - 1) = (n - 2) 	imes 180$.Dividing both sides by $60$: $2(2^n - 1) = 3(n - 2)$.$2^{n+1} - 2 = 3n - 6$,which simplifies to $2^{n+1} + 4 = 3n$.For $n=3$,$2^4 + 4 = 20$ and $3(3) = 9$ $(20 
eq 9)$.For $n=4$,$2^5 + 4 = 36$ and $3(4) = 12$ $(36 
eq 12)$.As $n$ increases,$2^{n+1}$ grows much faster than $3n$. Thus,there are no integer solutions for $n \ge 3$.

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $120^\circ$ and the common ratio is $2$. Find the number of possible values of $n$.

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