Answer
$F = G~\frac{m^2}{r^2}$
Note that this force is directed upward.
Work Step by Step
Note that all of the gravitational forces on the central particle cancel out except for the gravitational force from the particle located directly above the central particle on the ring of radius $r$.
We can find the magnitude of the net gravitational force on the central particle:
$F = G~\frac{m^2}{r^2}$
Note that this force is directed upward.
