| Exam Board | OCR MEI |
|---|---|
| Module | Further Extra Pure (Further Extra Pure) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Sequence Behaviour Classification |
| Difficulty | Standard +0.3 This question requires students to observe and describe patterns from numerical data without solving recurrence relations. While it involves Further Maths content (recurrence relations), the task is purely descriptive pattern recognition: identifying that a_n oscillates between 3 and -1, b_n converges to 2, and c_n diverges to infinity. No calculation, proof, or deep conceptual insight is required—just careful observation of the given spreadsheet data. |
| Spec | 1.04e Sequences: nth term and recurrence relations8.01a Recurrence relations: general sequences, closed form and recurrence8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states |
| A | B | C | D | |
| 1 | \(n\) | \(a _ { n }\) | \(b _ { n }\) | \(c _ { n }\) |
| 2 | 1 | 3 | 1.5 | 2.5 |
| 3 | 2 | -1 | 2.25 | 7.25 |
| 4 | 3 | 3 | 1.875 | 27.28125 |
| 5 | 4 | -1 | 2.0625 | 249.0889 |
| 6 | 5 | 3 | 1.96875 | 15512.32 |
| 7 | 6 | -1 | 2.01563 | 48126390 |
| 8 | 7 | 3 | 1.99219 | 3.86E+14 |
| 9 | 8 | -1 | 2.00391 | \(2.13 \mathrm { E } + 28\) |
| 10 | 9 | 3 | 1.99805 | 5.66E+55 |
| 11 | 10 | -1 | 2.00098 | 3.6E+110 |
| Answer | Marks |
|---|---|
| 1 | a : periodic... |
| Answer | Marks |
|---|---|
| ...divergent | B1 |
| Answer | Marks |
|---|---|
| [7] | 2.2b |
| Answer | Marks |
|---|---|
| 2.2b | eg “the smallest infinitely |
| Answer | Marks |
|---|---|
| limit” | If B0B0 for description of sequence |
Question 1:
1 | a : periodic...
n
with period 2 oe
b : oscillatory...
n
convergent...
...limit is 2 (or 2.0 or 2.00)
c : increasing...
n
...divergent | B1
B1
B1
B1
B1
B1
B1
[7] | 2.2b
2.2b
2.2b
2.2b
2.2b
2.2b
2.2b | eg “the smallest infinitely
repeating sub-sequence
containing two numbers”.
Not 1.9 or 2.0 or 1.9... or 2.0....
Must be a correct statement.
or “not convergent” or “tends to
infinity” or “(increases) without
limit” | If B0B0 for description of sequence
a then SC1 for stating either that
n
sequence a does not converge or
n
that it oscillates.1 Three sequences, $\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }$ and $\mathrm { c } _ { \mathrm { n } }$, are defined for $n \geqslant 1$ by the following recurrence relations.
$$\begin{aligned}
& \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3 \\
& b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5 \\
& c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5
\end{aligned}$$
The output from a spreadsheet which presents the first 10 terms of $a _ { n } , b _ { n }$ and $c _ { n }$, is shown below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& A & B & C & D \\
\hline
1 & $n$ & $a _ { n }$ & $b _ { n }$ & $c _ { n }$ \\
\hline
2 & 1 & 3 & 1.5 & 2.5 \\
\hline
3 & 2 & -1 & 2.25 & 7.25 \\
\hline
4 & 3 & 3 & 1.875 & 27.28125 \\
\hline
5 & 4 & -1 & 2.0625 & 249.0889 \\
\hline
6 & 5 & 3 & 1.96875 & 15512.32 \\
\hline
7 & 6 & -1 & 2.01563 & 48126390 \\
\hline
8 & 7 & 3 & 1.99219 & 3.86E+14 \\
\hline
9 & 8 & -1 & 2.00391 & $2.13 \mathrm { E } + 28$ \\
\hline
10 & 9 & 3 & 1.99805 & 5.66E+55 \\
\hline
11 & 10 & -1 & 2.00098 & 3.6E+110 \\
\hline
\end{tabular}
\end{center}
Without attempting to solve any recurrence relations, describe the apparent behaviour, including as $n \rightarrow \infty$, of
\begin{itemize}
\item $a _ { n }$
\item $\mathrm { b } _ { \mathrm { n } }$
\item $\mathrm { C } _ { \mathrm { n } }$
\end{itemize}
\hfill \mbox{\textit{OCR MEI Further Extra Pure 2022 Q1 [7]}}