1. **State the problem:** We have a geometric sequence defined by the first term $a_1 = 8$ and common ratio $r = \frac{3}{4}$. We need to find the sum of the first 25 terms, $S_{25}$. 2. **Formula for the sum of the first $n$ terms of a geometric sequence:** $$ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} $$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. 3. **Apply the formula:** $$ S_{25} = 8 \cdot \frac{1 - \left(\frac{3}{4}\right)^{25}}{1 - \frac{3}{4}} = 8 \cdot \frac{1 - \left(\frac{3}{4}\right)^{25}}{\frac{1}{4}} = 8 \cdot 4 \cdot \left(1 - \left(\frac{3}{4}\right)^{25}\right) $$ 4. **Simplify:** $$ S_{25} = 32 \cdot \left(1 - \left(\frac{3}{4}\right)^{25}\right) $$ 5. **Calculate $\left(\frac{3}{4}\right)^{25}$:** This is a very small number because $\left(\frac{3}{4}\right) < 1$ and raised to a large power. 6. **Approximate:** $$ \left(\frac{3}{4}\right)^{25} \approx 0.000177 $$ 7. **Calculate the sum:** $$ S_{25} \approx 32 \cdot (1 - 0.000177) = 32 \cdot 0.999823 = 31.994 $$ 8. **Final answer:** The sum of the first 25 terms is approximately $31.98$, which corresponds to option C.