Answer
The proof is below.
Work Step by Step
We know that the mass will be located at the average of the two radii. Thus, we find:
$I_{ring} = \frac{MR_1^2 + MR_2^2}{2}$
$I_{ring} = \frac{M(R_1^2 + R_2^2)}{2}$
$I_{ring} = \frac{1}{2}M(R_1^2 + R_2^2)$
