1. Capacitance of an isolated conducting sphere

For an isolated conducting sphere of radius $R_1$, the capacitance is

$C_1 = 4\pi \varepsilon_0 R_1$

2. Capacitance when enclosed by a concentric earthed conducting sphere

If the sphere of radius $R_1$ is enclosed by a concentric conducting sphere of radius $R_2$ connected to earth, the arrangement becomes a spherical capacitor.

Its capacitance is

$C_2 = 4\pi \varepsilon_0 \frac{R_1R_2}{R_2-R_1}$

3. Given condition

The capacitance becomes $n$ times:

$C_2 = n C_1$

Substitute the expressions:

$4\pi \varepsilon_0 \frac{R_1R_2}{R_2-R_1} = n \left(4\pi \varepsilon_0 R_1\right)$

Cancel $4\pi \varepsilon_0 R_1$ from both sides:

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

4. Solve for $\dfrac{R_2}{R_1}$

$R_2 = n(R_2-R_1)$

$R_2 = nR_2 - nR_1$

$nR_1 = (n-1)R_2$

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

5. Check options

This matches Option A.

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