1. **State the problem:** We have a geometric sequence defined by the first term $a_1 = 8$ and the common ratio $r = -1.5$. We want to find the sum of the first 20 terms, denoted as $S_{20}$. 2. **Formula for the sum of the first $n$ terms of a geometric sequence:** $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. 3. **Apply the formula:** Given $a_1 = 8$, $r = -1.5$, and $n = 20$, substitute these values: $$S_{20} = 8 \frac{1 - (-1.5)^{20}}{1 - (-1.5)}$$ 4. **Calculate the denominator:** $$1 - (-1.5) = 1 + 1.5 = 2.5$$ 5. **Calculate the numerator:** Calculate $(-1.5)^{20}$. Since 20 is even, $(-1.5)^{20} = (1.5)^{20}$, which is a positive number. 6. **Evaluate $(1.5)^{20}$:** Using a calculator or approximation, $(1.5)^{20} \approx 33252.16$ 7. **Calculate numerator:** $$1 - 33252.16 = -33251.16$$ 8. **Calculate the sum:** $$S_{20} = 8 \times \frac{-33251.16}{2.5} = 8 \times (-13300.464) = -106403.712$$ **Final answer:** $$\boxed{-106403.712}$$ This means the sum of the first 20 terms of the sequence is approximately $-106403.712$.