Given a G.P. with a = 729 and 7th term 64, determine S7.

$a=729$

$a_{7}=64$

Let $r$ be the common ratio of the G.P.

It is known that, $a_{n}=a r^{n-1}$

$a_{7}=a r^{7-1}=(729) r^{6}$

$\Rightarrow 64=729 r^{6}$

$\Rightarrow r^{6}=\frac{64}{729}$

$\Rightarrow r^{6}=\left(\frac{2}{3}\right)^{6}$

$\Rightarrow r=\frac{2}{3}$

Also, it is known that, $S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$

$\therefore S_{7}=\frac{729\left[1-\left(\frac{2}{3}\right)^{7}\right]}{1-\frac{2}{3}}$

$=3 \times 729\left[1-\left(\frac{2}{3}\right)^{7}\right]$

$=(3)^{7}\left[\frac{(3)^{7}-(2)^{7}}{(3)^{7}}\right]$

$=(3)^{7}-(2)^{7}$

$=2187-128$

$=2059$