Kolmogorov complexity of prefixes of computable sequences
Let $D$ be a decider for $A$, then in order to determine $\mathcal{X}_{A,n}$ it is enough to specify $n$ and an encoding of $D$. Given $D,n$, you can loop over all integers $i\in [n]$ and check whether or not $i\in A$ by simulating $D(i)$. Thus, the overall description of $\mathcal{X}_{A,n}$ consists of $n$, the encoding of $D$ and the encapsulating machine $M$ (described above) which uses $D,n$ to output $\mathcal{X}_{A,n}$. $n$ contributes at most $\log n$ bits to the description, and $D,M$ have some encoding of length $c$ independent of $n$, bounding the overall description length by $\log n + c$.