Geometric Sum 3Db6Ad

1. **State the problem:** We need to find the sum of the first 10 terms of the geometric sequence $$\{1.5, 3, 6, 12, 24, ...\}$$. 2. **Identify the first term and common ratio:** The first term $$a = 1.5$$. The common ratio $$r = \frac{3}{1.5} = 2$$. 3. **Formula for the sum of the first $$n$$ terms of a geometric sequence:** $$ S_n = a \frac{r^n - 1}{r - 1} $$ where $$S_n$$ is the sum of the first $$n$$ terms, $$a$$ is the first term, $$r$$ is the common ratio, and $$n$$ is the number of terms. 4. **Apply the formula for $$n=10$$:** $$ S_{10} = 1.5 \times \frac{2^{10} - 1}{2 - 1} $$ 5. **Calculate $$2^{10}$$:** $$ 2^{10} = 1024 $$ 6. **Substitute and simplify:** $$ S_{10} = 1.5 \times \frac{1024 - 1}{1} = 1.5 \times 1023 $$ 7. **Multiply:** $$ S_{10} = 1.5 \times 1023 = 1534.5 $$ **Final answer:** The sum of the first 10 terms is $$\boxed{1534.5}$$.