Given a $G.P.$ with $a=729$ and $7^{\text {th }}$ term $64,$ determine $S_{7}$
Given a $G.P.$ with $a=729$ and $7^{\text {th }}$ term $64,$ determine $S_{7}$
$a=729 a_{7}=64$
Let $r$ be the common ratio of the $G.P.$ It is known that,
$a_{n}=a r^{n-1}$
$a_{7}=a r^{7-1}=(729) r^{6}$
$\Rightarrow 64=729 r^{6}$
$\Rightarrow r^{6}=\left(\frac{2}{3}\right)^{6}$
$\Rightarrow r=\frac{2}{3}$
Also, it is known that,
$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$
$\therefore S_{7}=\frac{729\left(1-\left(\frac{2}{3}\right)^{7}\right)}{1-\frac{2}{3}}$
$=3 \times 729\left[1-\left(\frac{2}{3}\right)^{7}\right]$
$=(3)^{7}\left[\frac{(3)^{7}-(2)^{7}}{(3)^{7}}\right]$
$=(3)^{7}-(2)^{7}$
$=2187-128$
$=2059$
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