### Instruction: Kevin Kangaroo begins hopping on a number line at 0. He wants to get to 1, but he can hop only $\frac{1}{3}$ of the distance. Each hop tires him out so that he continues to hop $\frac{1}{3}$ of the remaining distance. How far has he hopped after five hops? Express your answer as a common fraction.### Response: I know that the formula for the sum of a geometric series is $S_n = a \frac{1 - r^n}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
In this problem, the first term is $\frac{1}{3}$, the common ratio is also $\frac{1}{3}$, and the number of terms is 5.
So I can plug these values into the formula and simplify: $S_5 = \frac{1}{3} \frac{1 - (\frac{1}{3})^5}{1 - \frac{1}{3}} = \frac{1}{3} \frac{1 - \frac{1}{243}}{\frac{2}{3}} = \frac{1}{2} \frac{242}{243} = \frac{121}{243}$.