Suppose the gravitational force varies inversely as the nth power of distance then the time period T of a satellite revolving in a circular orbit of radius r around the earth is proportional to

$r^{\frac{n+1}{2}}$

$\frac{G M m}{r^n}=m r\left(\frac{2 \pi}{T}\right)^2$

$T^2=\frac{4 \pi^2 r^{n+1}}{G M} \Rightarrow T \propto r^{\frac{n+1}{2}}$