### Instruction: Find the sum of the first six terms in the geometric sequence $\frac12,\frac14,\frac18,\dots$. Express your answer as a common fraction.### Response: I recognize that this is a geometric sequence, where each term is obtained by multiplying the previous term by a common ratio.
In this case, the common ratio is $\frac12$, since $\frac14=\frac12 \cdot \frac12$, $\frac18=\frac14 \cdot \frac12$, and so on.
To find the sum of the first six terms, I can use the formula for the sum of a finite geometric series, which is $S_n=a_1\cdot \frac{1-r^n}{1-r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
In this problem, $a_1=\frac12$, $r=\frac12$, and $n=6$, so I can plug these values into the formula and simplify.