Given law of gravitation

The gravitational force is said to vary inversely as the $n$th power of distance: $F \propto \frac{1}{R^n}$

For circular motion

A planet of mass $m$ moving in a circular orbit of radius $R$ requires centripetal force: $F_c = \frac{mv^2}{R}$

This centripetal force is provided by gravity, so $\frac{mv^2}{R} \propto \frac{1}{R^n}$

Find the speed dependence on $R$

Multiplying by $R$: $mv^2 \propto \frac{1}{R^{n-1}}$ $v^2 \propto R^{-(n-1)}$ $v \propto R^{-\frac{n-1}{2}}$

Time period of revolution

The time period is $T = \frac{2\pi R}{v}$

Substituting the proportionality for $v$: $T \propto R \cdot R^{\frac{n-1}{2}}$ $T \propto R^{1 + \frac{n-1}{2}}$ $T \propto R^{\frac{n+1}{2}}$

Match with the options

Therefore, $T \propto R^{\frac{n+1}{2}}$

This corresponds to Option C.

Verification with stored answer

Stored correct answer: C

Derived answer: C

So, the derived answer agrees with the stored answer.