| Exam Board | Edexcel |
|---|---|
| Module | FD2 AS (Further Decision 2 AS) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Applied recurrence modeling |
| Difficulty | Standard +0.8 This is a Further Maths question requiring pattern recognition to establish a recurrence relation (u_{n+1} = u_n + n + 1), then solving it to get u_n = 1 + n(n+1)/2. While the pattern-spotting and solving first-order recurrence is accessible, it requires genuine mathematical insight rather than routine application, and the context is non-standard for most students. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| 16, 22, 29 | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_{n+1} = u_n + n + 1\) | B1 | Translating problem to mathematical model - correct recurrence relation needed |
| Answer | Marks | Guidance |
|---|---|---|
| As \(u_{n+1} = u_n + p(n) \Rightarrow u_n = \lambda n^2 + \mu n + \phi\), attempt to solve with \(n = 1, 2, 3\) | M1 | An attempt to solve the recurrence relation to determine maximum number of regions |
| \(u_n = \frac{1}{2}n(n+1)+1\) | A1 | cao |
| 20 101 (regions) | A1ft | Substitution of \(n = 200\) into their quadratic \(u_n\) expression |
## Question 2:
### Part (a):
| 16, 22, 29 | B1 | cao |
### Part (b):
| $u_{n+1} = u_n + n + 1$ | B1 | Translating problem to mathematical model - correct recurrence relation needed |
### Part (c):
| As $u_{n+1} = u_n + p(n) \Rightarrow u_n = \lambda n^2 + \mu n + \phi$, attempt to solve with $n = 1, 2, 3$ | M1 | An attempt to solve the recurrence relation to determine maximum number of regions |
| $u_n = \frac{1}{2}n(n+1)+1$ | A1 | cao |
| 20 101 (regions) | A1ft | Substitution of $n = 200$ into their quadratic $u_n$ expression |
---2. In two-dimensional space, lines divide a plane into a number of different regions.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_421_328_306_278}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_423_330_306_671}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_426_330_303_1065}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{81510c1c-89ce-4fa8-aa1b-3c8b255804cc-3_423_332_306_1457}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
It is known that:
\begin{itemize}
\item One line divides a plane into 2 regions, as shown in Figure 1
\item Two lines divide a plane into a maximum of 4 regions, as shown in Figure 2
\item Three lines divide a plane into a maximum of 7 regions, as shown in Figure 3
\item Four lines divide a plane into a maximum of 11 regions, as shown in Figure 4
\begin{enumerate}[label=(\alph*)]
\item Complete the table in the answer book to show the maximum number of regions when five, six and seven lines divide a plane.
\item 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 $n \geqslant 1$
\item \begin{enumerate}[label=(\roman*)]
\item Solve the recurrence relation for $u _ { n }$
\item Hence determine the maximum number of regions created when 200 lines divide a plane.
\end{itemize}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 AS Q2 [5]}}