| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Periodic Sequences |
| Difficulty | Standard +0.8 Part (i) is a routine geometric series calculation (below average difficulty). However, part (ii) requires students to discover a periodic pattern by computing terms of a recursively defined sequence, recognize the period, and then use this to sum 70 terms—this involves non-standard problem-solving and pattern recognition beyond typical A-level questions. The combination elevates this to moderately above average difficulty. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r = 0.25\) | B1 | May be implied by use in formula; \(6\times 0.25\) alone is not sufficient evidence of geometric sequence |
| \(S_\infty = \frac{1.5}{1-0.25} = 2\) | M1 | Attempts \(S_\infty = \frac{a}{1-r}\) with \(a=1.5\), \(r=0.25\) or \(a=6\), \(r=0.25\) |
| \(= 2\) | A1 | cao; must follow evidence of sum to infinity formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((u_1=3),\ u_2=0,\ u_3=\frac{3}{2},\ u_4=3, \ldots\) and states \(u_4=u_1\) so sequence repeats (is periodic) | M1 | Uses recurrence relation correctly at least once; if \(u_2\neq 0\) check \(u_3\) is correct follow-through |
| Full sequence \((u_1=3),\ u_2=0,\ u_3=\frac{3}{2},\ u_4=3,\ldots\) with conclusion that sequence is periodic | A1 | Minimum accept \((3),\ 0,\ \frac{3}{2},\ 3\) (hence periodic) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (Order) 3 | A1 | Do not accept "repeats every third"; cannot be scored for just listing terms; can only be scored following correct values in (ii)(a) or a restart |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum_{n=1}^{70} u_n = 23\times\left(3+0+\frac{3}{2}\right)+3 = 106.5\) | M1 | Establishes correct method for periodic sequence of order 3 found in (a); attempts to use geometric/arithmetic series is M0A0 |
| \(= 106.5\) | A1 | Or exact equivalent e.g. \(\frac{213}{2}\); ignore subsequent incorrect rounding |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = 0.25$ | B1 | May be implied by use in formula; $6\times 0.25$ alone is not sufficient evidence of geometric sequence |
| $S_\infty = \frac{1.5}{1-0.25} = 2$ | M1 | Attempts $S_\infty = \frac{a}{1-r}$ with $a=1.5$, $r=0.25$ or $a=6$, $r=0.25$ |
| $= 2$ | A1 | cao; must follow evidence of sum to infinity formula |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(u_1=3),\ u_2=0,\ u_3=\frac{3}{2},\ u_4=3, \ldots$ and states $u_4=u_1$ so sequence repeats (is periodic) | M1 | Uses recurrence relation correctly at least once; if $u_2\neq 0$ check $u_3$ is correct follow-through |
| Full sequence $(u_1=3),\ u_2=0,\ u_3=\frac{3}{2},\ u_4=3,\ldots$ with conclusion that sequence is periodic | A1 | Minimum accept $(3),\ 0,\ \frac{3}{2},\ 3$ (hence periodic) |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (Order) 3 | A1 | Do not accept "repeats every third"; cannot be scored for just listing terms; can only be scored following correct values in (ii)(a) or a restart |
### Part (ii)(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum_{n=1}^{70} u_n = 23\times\left(3+0+\frac{3}{2}\right)+3 = 106.5$ | M1 | Establishes correct method for periodic sequence of order 3 found in (a); attempts to use geometric/arithmetic series is M0A0 |
| $= 106.5$ | A1 | Or exact equivalent e.g. $\frac{213}{2}$; ignore subsequent incorrect rounding |\begin{enumerate}
\item (i) Find the value of
\end{enumerate}
$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$
(3)\\
(ii) A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
$$\begin{aligned}
u _ { 1 } & = 3 \\
u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N }
\end{aligned}$$
(a) Show that this sequence is periodic.\\
(b) State the order of this sequence.\\
(c) Hence find
$$\sum _ { n = 1 } ^ { 70 } u _ { n }$$
\hfill \mbox{\textit{Edexcel P2 2024 Q5 [8]}}