Can someone tell me the formula for finding the sum of a finite qeometric series
let $\displaystyle S=\sum\limits_{i=1}^{n}ar^{i-1}$
then
$\displaystyle S=a+ar+ar^{2}+\cdots +ar^{n-1}$
and
$\displaystyle rS=ar+ar^{2}+ar^{3}+\cdots +ar^{n}$
substract this two equations yield
$\displaystyle S(1-r)=a-ar^{n}$
$\displaystyle S=\frac{a-ar^{n}}{(1-r)}$
look at the formula,.
the first term $\displaystyle a=1$, the ratio $\displaystyle r=2/3$, and
the last term is $\displaystyle 64/729$
so
$\displaystyle ar^{n-1}=64/729$
$\displaystyle \left(\frac{2}{3}\right)^{(n-1)}=\frac{64}{729}=\left(\frac{2}{3}\right)^6$
$\displaystyle n-1=6$
$\displaystyle n=7$