2. In two-dimensional space, lines divide a plane into a number of different regions.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_421_328_306_278}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_423_330_306_671}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_426_330_303_1065}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_423_332_306_1457}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
It is known that:
- One line divides a plane into 2 regions, as shown in Figure 1
- Two lines divide a plane into a maximum of 4 regions, as shown in Figure 2
- Three lines divide a plane into a maximum of 7 regions, as shown in Figure 3
- Four lines divide a plane into a maximum of 11 regions, as shown in Figure 4
- Complete the table in the answer book to show the maximum number of regions when five, six and seven lines divide a plane.
- Find, in terms of \(\mathrm { u } _ { \mathrm { n } }\), the recurrence relation for \(\mathrm { u } _ { \mathrm { n } + 1 }\), the maximum number of regions when a plane is divided by ( \(n + 1\) ) lines where \(n \geqslant 1\)
- Solve the recurrence relation for \(u _ { n }\)
- Hence determine the maximum number of regions created when 200 lines divide a plane.