| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2014 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Periodic or repeating sequence |
| Difficulty | Standard +0.8 This recurrence relation question requires students to discover a periodic pattern (cycle of 3 values) through calculation, then apply this insight to find distant terms and sum 99 terms. While the arithmetic is straightforward, recognizing the periodicity and using it strategically for parts (b) and (c) requires problem-solving beyond routine sequence exercises, making it moderately challenging for C1/C2 level. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u_2 = 2 - \frac{4}{\frac{2}{3}} = \frac{2}{3}\), \(u_3 = 2 - \frac{4}{\frac{2}{3}} = -4\), \(u_4 = 2 - \frac{4}{-4} = 3\) | M1 A1 A1 | M1: Attempt to use formula correctly. A1: two correct answers. A1: 3 correct answers (allow \(0.\overline{6}\) but not \(0.667\)) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u_{61} = 3\) | B1 | B1: cao. NB Use of AP is B0 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum_{i=1}^{99} u_i = (3 + \frac{2}{3} - 4) + (3 + \frac{2}{3} - 4) + \ldots\) | M1 | M1: Uses sum of at least 3 terms found from part (a). Attempt to sum an AP is M0 |
| \(\sum_{i=1}^{99} u_i = 33 \times (\ldots + \ldots + \ldots)\), \(= -11\) | A1, A1 | A1: obtains \(33 \times\) (sum of three adjacent terms). A1: \(-11\) cao. N.B. Use of \(n=99\) is M1A0A0 |
| [3] |
# Question 5:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u_2 = 2 - \frac{4}{\frac{2}{3}} = \frac{2}{3}$, $u_3 = 2 - \frac{4}{\frac{2}{3}} = -4$, $u_4 = 2 - \frac{4}{-4} = 3$ | M1 A1 A1 | M1: Attempt to use formula correctly. A1: two correct answers. A1: 3 correct answers (allow $0.\overline{6}$ but not $0.667$) |
| | [3] | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u_{61} = 3$ | B1 | B1: cao. NB Use of AP is B0 |
| | [1] | |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum_{i=1}^{99} u_i = (3 + \frac{2}{3} - 4) + (3 + \frac{2}{3} - 4) + \ldots$ | M1 | M1: Uses sum of at least 3 terms found from part (a). Attempt to sum an AP is M0 |
| $\sum_{i=1}^{99} u_i = 33 \times (\ldots + \ldots + \ldots)$, $= -11$ | A1, A1 | A1: obtains $33 \times$ (sum of three adjacent terms). A1: $-11$ cao. N.B. Use of $n=99$ is M1A0A0 |
| | [3] | |
---5. A sequence is defined by
$$\begin{aligned}
u _ { 1 } & = 3 \\
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1
\end{aligned}$$
Find the exact values of
\begin{enumerate}[label=(\alph*)]
\item $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$
\item $u _ { 61 }$
\item $\sum _ { i = 1 } ^ { 99 } u _ { i }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2014 Q5 [7]}}