= Solution
First suppose $X$ is integral. Induct on its <dimension of a scheme>. The previous two parts handle all $j>1$ and also handle $j=1$ when $h^0(X,mD)$ eventually vanishes. Otherwise choose $r>0$ with a nonzero <global section> of $rD$. On an integral variety it defines an <effective Cartier divisor> $E$, possibly empty, and multiplication by the section gives
$$
0\to\mathcal O_X((m-r)D)\to\mathcal O_X(mD)\to\mathcal O_E(mD)\to0.
$$
Thus $h^1(X,mD)\leq h^1(X,(m-r)D)+h^1(E,mD)$. For $n\geq2$, the induction hypothesis bounds the last term by $O(m^{n-2})$; summing on each residue class modulo $r$ gives $O(m^{n-1})$. For $n=1$, $E$ is zero-dimensional and its positive-degree <sheaf cohomology> vanishes, so the same recurrence is bounded.
For a general <projective scheme>, a nonzero section can be a <zero divisor>, so that argument requires an additional step. The general <cohomology growth for nef twists> supplies it: for every <coherent sheaf> $\mathcal F$ with support dimension $d$,
$$
h^i(X,\mathcal F\otimes\mathcal O_X(mD))=O(m^{d-i})\quad(1\leq i\leq d).
$$
Its proof uses <Fujita vanishing>, an ample section avoiding the <associated points> of $\mathcal F$, and induction on support dimension. Taking $\mathcal F=\mathcal O_X$ gives the required $O(m^{n-1})$ estimate for every $j\geq1$, including nonreduced and reducible schemes; degrees above $n$ vanish.
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