Answer
$\displaystyle \frac{80}{81}$
Work Step by Step
See p. 861.
For the geometric sequence $a_{n}=ar^{n-1}$
the nth partial sum$ S_{n}=\displaystyle \sum_{k=1}^{n}ar^{k-1}$ (where $r\neq 1$)
is given by$ \displaystyle \quad S_{n}=a\cdot\frac{1-r^{n}}{1-r}$
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$S_{4}=(\displaystyle \frac{2}{3})\frac{1-(\frac{1}{3})^{4}}{1-\frac{1}{3}}$
$=\displaystyle \frac{2}{3}\times\frac{\frac{80}{81}}{\frac{2}{3}}$
$=\displaystyle \frac{2}{3}\times\frac{80}{81}\times\frac{3}{2}$
$=\displaystyle \frac{80}{81}$.
