= Solution
Choose a spherical <Gaussian surface> of radius $s$ with $a<s<r_0$. Its physical enclosed <electric charge> is the total induced <electric charge> $Q_a$ on the inner sphere. The <method of images> produces the same <electric field> on that <Gaussian surface>, so <Gauss's law> identifies $Q_a$ with the sum of the enclosed <image charges>. These are the first family with $n\leq-1$ and the second family with $n\leq0$:
$$
Q_a=q\sum_{n=-\infty}^{-1}k^n-\frac{qa}{r_0}\sum_{n=-\infty}^{0}k^n
=q\frac{k^{-1}-a/r_0}{1-k^{-1}}.
$$
Summing these <geometric series> gives
$$
\boxed{Q_a=-q\frac{a(b-r_0)}{r_0(b-a)}.}
$$
The <electric field> in the empty central cavity vanishes by <electrostatic shielding>, so it contributes no additional inner-face <surface charge density>. The formula tends to $-q$ as $r_0\downarrow a$ and to zero as $r_0\uparrow b$, as expected when the <point charge> approaches one grounded boundary or the other.
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