1. The problem asks to find the geometric means of the sequence: 0.5, ____, ____, ____, 2048. 2. The geometric means are the terms between the first and last term in a geometric sequence. 3. The formula for the $n$th term of a geometric sequence is $$a_n = a_1 \times r^{n-1}$$ where $a_1$ is the first term and $r$ is the common ratio. 4. Here, the sequence has 5 terms: $a_1 = 0.5$, $a_5 = 2048$, and we need to find $a_2$, $a_3$, and $a_4$. 5. Using the formula for the 5th term: $$a_5 = a_1 \times r^{4}$$ 6. Substitute known values: $$2048 = 0.5 \times r^{4}$$ 7. Divide both sides by 0.5: $$\frac{2048}{\cancel{0.5}} = \frac{0.5 \times r^{4}}{\cancel{0.5}} \Rightarrow 4096 = r^{4}$$ 8. Take the 4th root of both sides to find $r$: $$r = \sqrt[4]{4096}$$ 9. Since $4096 = 2^{12}$, then $$r = 2^{\frac{12}{4}} = 2^{3} = 8$$ 10. Now find the missing terms: - $$a_2 = a_1 \times r = 0.5 \times 8 = 4$$ - $$a_3 = a_1 \times r^{2} = 0.5 \times 8^{2} = 0.5 \times 64 = 32$$ - $$a_4 = a_1 \times r^{3} = 0.5 \times 8^{3} = 0.5 \times 512 = 256$$ 11. The geometric means are 4, 32, and 256. 12. Comparing with the multiple-choice answers, the correct choice is J: 4, 32, 256.