Find the sum of the first five terms in the geometric sequence $\frac13,\frac19,\frac1{27},\dots$. Express your answer as a common fraction.
This is a finite geometric series with first term $\frac13$ and common ratio $\frac13$. There are five terms, so the sum of this series is $\frac{\frac13\left(1-\left(\frac13\right)^5\right)}{1-\frac13} = \boxed{\frac{121}{243}}$.