1. **State the problem:** Find the sum of the first five terms of a geometric series where the first term $a=3$ and the common ratio $r=2$. 2. **Formula used:** The sum of the first $n$ terms of a geometric series is given by $$S_n = a \frac{r^n - 1}{r - 1}$$ where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. 3. **Substitute the values:** Here, $a=3$, $r=2$, and $n=5$, so $$S_5 = 3 \frac{2^5 - 1}{2 - 1}$$ 4. **Calculate powers and simplify:** $$2^5 = 32$$ So, $$S_5 = 3 \frac{32 - 1}{1} = 3 \times 31$$ 5. **Final calculation:** $$S_5 = 93$$ **Answer:** The sum of the first five terms is $93$.