Example 4. On a plane, there are $\mathrm{n}$ lines, among which no two are parallel, and no three or more pass through the same point. How many regions does the plane get divided into by these $\mathrm{n}$ lines?
Solve by first exploring the pattern using $\mathrm{n}=1,2,3,4$, etc.

Let the number of regions into which the plane is divided by $\mathrm{n}$ lines be $\mathrm{P}_{\mathrm{n}}$. From the diagram, we can see that
$$
\begin{array}{l}
\mathrm{P}_{1}=2, \\
\mathrm{P}_{2}=4=2+2=\mathrm{P}_{1}+2, \\
\mathrm{P}_{3}=7=4+3=\mathrm{P}_{2}+3, \\
\mathrm{P}_{4}=11=7+4=\mathrm{P}_{3}+4, \\
\cdots \cdots
\end{array}
$$
$$
P_{n-1}=P_{n-2}+n-1,
$$
$$
\begin{aligned}
+) \mathrm{P}_{n} & =\mathrm{P}_{n-1}+\mathrm{n} \\
\mathrm{P}_{n} & =2+(2+3+4+\cdots+\mathrm{n}) \\
& =1+\frac{\mathrm{n}(1+\mathrm{n})}{2}
\end{aligned}
$$