Capacitance of an isolated conducting sphere of radius $$R_1$$ becomes $$n$$ times when it is enclosed by a concentric conducting sphere of radius $$R_2$$ connected to earth. The ratio of their radii  $$\left(\dfrac{R_2}{R_1}\right)$$ is:

We need to find the ratio $$\frac{R_2}{R_1}$$ when the capacitance of an isolated sphere becomes $$n$$ times after enclosing it with a grounded concentric sphere.

Write the capacitance of an isolated sphere.

$$C_1 = 4\pi\varepsilon_0 R_1$$

Write the capacitance of the spherical capacitor.

When the inner sphere of radius $$R_1$$ is enclosed by a grounded outer sphere of radius $$R_2$$, the capacitance is:

$$C_2 = \frac{4\pi\varepsilon_0 R_1 R_2}{R_2 - R_1}$$

Apply the condition $$C_2 = nC_1$$.

$$\frac{4\pi\varepsilon_0 R_1 R_2}{R_2 - R_1} = n \times 4\pi\varepsilon_0 R_1$$

$$\frac{R_2}{R_2 - R_1} = n$$

$$R_2 = n(R_2 - R_1) = nR_2 - nR_1$$

$$nR_1 = nR_2 - R_2 = R_2(n - 1)$$

$$\frac{R_2}{R_1} = \frac{n}{n - 1}$$

The correct answer is Option A: $$\dfrac{n}{n-1}$$.