Example: $8 n$ planes can divide the space into at most how many regions?
To find this maximum value, we first prove the following lemma: $n$ lines in a plane can divide the plane into at most $\mathrm{C}_{n+1}^{2}+1$ parts.

Obviously, when these $n$ lines intersect each other pairwise and no three lines intersect at the same point, the number of parts the plane is divided into is maximized.

Let the maximum number of parts the plane is divided into by $k$ lines be $m_{k}$. Then, add the $(k+1)$-th line, which intersects with each of the previous $k$ lines, resulting in $k$ intersection points. These $k$ points divide the $(k+1)$-th line into $k+1$ segments, each of which divides the region it passes through into two parts. Therefore, the number of new small regions added due to the addition of the $(k+1)$-th line is $k+1$. Thus, we obtain the recursive formula
$$
m_{k+1}=m_{k}+k+1 \text {, }
$$

From this recursion, we get
$$
\begin{aligned}
m_{n} & =m_{n-1}+n=n+n-1+m_{n-2} \\
= & n+n-1+\cdots+2+m_{1}=n+n-1+\cdots+ \\
2+1 & +1=\mathrm{C}_{n+1}^{1}+1 .
\end{aligned}
$$

This completes the proof of the lemma. Now, we use the lemma to solve the original problem.

Let the maximum number of regions that $k$ planes can divide the space into be $v_{k}$. Then, consider the number of new small regions added when the $(k+1)$-th plane is added. At this point, the $(k+1)$-th plane intersects each of the previous $k$ planes along one line, resulting in $k$ lines on the $(k+1)$-th plane. By the lemma, these $k$ lines can divide the plane into at most $\mathrm{C}_{k+1}^{2}+1$ parts, each of which divides the region it passes through into two parts. Therefore, we get the recursive relation
$$
v_{k+1}=v_{k}+m_{k},
$$

From this recursion, we get
$$
\begin{array}{l}
v_{n}=m_{n-1}+m_{n-2}+\cdots+m_{1}+v_{1} \\
=\mathrm{C}_{n}^{2}+\mathrm{C}_{n-1}^{2}+\cdots+\mathrm{C}_{2}^{2}+(n-1)+2 \\
=\mathrm{C}_{n+1}^{3}+n+1,
\end{array}
$$

Thus, $n$ planes in space can divide the space into at most $\mathrm{C}_{n+1}^{3}+n+1$ parts, and this maximum value is achieved when any three planes intersect at a single point and no four planes intersect at the same point.