If the first term of a G.P. is 729 and 7^{th} | vedclass

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(C) Given that the first term $a = 729$ and the $7^{th}$ term $a_{7} = 64$.Let $r$ be the common ratio of the $G.P.$The formula for the $n^{th}$ term

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$2059$

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(C) Given that the first term $a = 729$ and the $7^{th}$ term $a_{7} = 64$.Let $r$ be the common ratio of the $G.P.$The formula for the $n^{th}$ term is $a_{n} = a r^{n-1}$.For $n = 7$,$a_{7} = a r^{6} = 64$.Substituting $a = 729$,we get $729 r^{6} = 64$.$r^{6} = \frac{64}{729} = \left(\frac{2}{3}ight)^{6}$.Thus,$r = \frac{2}{3}$.The sum of the first $n$ terms of a $G.P.$ is given by $S_{n} = \frac{a(1 - r^{n})}{1 - r}$.For $n = 7$,$S_{7} = \frac{729(1 - (2/3)^{7})}{1 - 2/3}$.$S_{7} = \frac{729(1 - 128/2187)}{1/3} = 3 	imes 729 	imes \left(\frac{2187 - 128}{2187}ight)$.$S_{7} = 2187 	imes \frac{2059}{2187} = 2059$.

If the first term of a $G.P.$ is $729$ and $7^{th}$ term is $64,$ determine $S_{7}$.

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