Example 2-44 How many regions does $n$ lines divide the plane into? Assume no $=$ lines intersect at a common point, and each pair of lines intersects.
Let $n$ lines divide the plane into $D_{n}$ regions. Then the $n$-th line is divided into $n$ segments by the other $n-1$ lines. These $n$ segments are exactly the boundaries of the $n$ new regions added.
So
$$
\begin{array}{l}
D_{n}=D_{n-1}+n \\
D_{1}=2
\end{array}
$$

As shown in Figure 2-11.
Also,
$D_{1}=D_{0}+1$, then $D_{0}=1$
$$
\begin{aligned}
D_{n} & =D_{n-1}+n \\
& =D_{n-2}+(n-1)+n \\
& =D_{n-3}+(n-2)+(n-1)+n \\
& =\cdots=D_{0}+n+(n-1)+\cdots+1 \\
& =1+\frac{1}{2} n(n+1)
\end{aligned}
$$