= Solution
Put $\beta=1-2/n$. The fixed point of this <affine recurrence> is $n/2$. Subtracting it gives $r_{k+1}-n/2=\beta(r_k-n/2)$. Repeated substitution, or <mathematical induction>, therefore yields
$$
\boxed{r_k=\frac n2+\frac n2\left(1-\frac2n\right)^k.}
$$
For $n>2$ this expression applies to every $k\ge0$. For $n=2$, state the initial value $r_0=2$ separately and obtain $r_k=1$ for every $k\ge1$, avoiding any need to interpret $0^0$. The expected red imbalance decays towards zero, although the urn composition itself remains random.
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