The mean proportional between $\dfrac{1}{2}$ and 128 is

1. 64
2. 32
3. 16
4. 8

Let the mean proportional be x.

Hence, the numbers are $\dfrac{1}{2}$, x and 128 are in continued in proportion.

$\therefore \dfrac{1}{2} : x : : x : 128 \\[0.5em] \Rightarrow \dfrac{\dfrac{1}{2}}{x} = \dfrac{x}{128} \\[0.5em] \Rightarrow x^2 = \dfrac{1}{2} \times 128 \\[0.5em] \Rightarrow x^2 = 64 \\[0.5em] \Rightarrow x = \sqrt{64} \\[0.5em] \Rightarrow x = 8.$

∴ Option 4 is the correct option.