= Solution
Choose a finite affine cover $\mathcal U=(U_j)$ of the <Noetherian scheme> $X$. Because $X$ is <separated scheme>[separated], every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the <closed immersion> is also affine. By the definition of the <direct image sheaf>,
$$
(i_*\mathcal F)(U_J)=\mathcal F(V_J).
$$
The <Čech cochain complex>[Čech complexes] for $i_*\mathcal F$ on $\mathcal U$ and for $\mathcal F$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant <quasi-coherent sheaf>[quasi-coherent sheaves], so the <acyclic cover theorem> gives
$$
H^q(X,i_*\mathcal F)\cong H^q(Z,\mathcal F)
$$
for every $q$. This is <cohomology under a closed immersion>.
For $X=\mathbb P_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The <cohomological dimension bound from an affine cover> therefore yields
$$
H^q(Z,\mathcal F)=0\qquad(q>n).
$$
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