| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.8 This is a straightforward recurrence relation question requiring systematic calculation of terms, pattern recognition, and simple algebraic manipulation. While it involves fractions and a general formula, the steps are routine and mechanical with no conceptual challenges beyond basic substitution—easier than average A-level questions. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a_1 = 4 \Rightarrow a_2 = \frac{4}{4+1}\) | M1 | Attempts recurrence relation correctly at least once |
| \(\frac{4}{5}, \frac{4}{9}, \frac{4}{13}\) | A1A1 | Two of \(\frac{4}{5}, \frac{4}{9}, \frac{4}{13}\) (first A1); all three correct (second A1). Allow 0.8 for \(\frac{4}{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(p = 4\), e.g. \(4 = \frac{4}{p+q}\), \(\frac{4}{5} = \frac{4}{2p+q}\) | M1 | \(a_n = \frac{4}{4n \pm \ldots}\) or uses 2 terms to set up and solve two correct equations for fractions in \(p\) and \(q\) |
| \(a_n = \frac{4}{4n-3} \Rightarrow p = 4\) and \(q = -3\) | A1 | Either \(a_n = \frac{4}{4n-3}\) OR \(p=4\) and \(q=-3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{4}{4N-3} = \frac{4}{321} \Rightarrow N = \ldots\) | M1 | Solves their \(\frac{4}{pN+q} = \frac{4}{321}\) to obtain a value for \(N\) |
| \((N =) 81\) | A1 | Cao (ignore what they use for \(N\)) |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_1 = 4 \Rightarrow a_2 = \frac{4}{4+1}$ | M1 | Attempts recurrence relation correctly at least once |
| $\frac{4}{5}, \frac{4}{9}, \frac{4}{13}$ | A1A1 | Two of $\frac{4}{5}, \frac{4}{9}, \frac{4}{13}$ (first A1); all three correct (second A1). Allow 0.8 for $\frac{4}{5}$ |
## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $p = 4$, e.g. $4 = \frac{4}{p+q}$, $\frac{4}{5} = \frac{4}{2p+q}$ | M1 | $a_n = \frac{4}{4n \pm \ldots}$ or uses 2 terms to set up and solve two correct equations for fractions in $p$ and $q$ |
| $a_n = \frac{4}{4n-3} \Rightarrow p = 4$ and $q = -3$ | A1 | Either $a_n = \frac{4}{4n-3}$ OR $p=4$ and $q=-3$ |
## Question 6(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{4}{4N-3} = \frac{4}{321} \Rightarrow N = \ldots$ | M1 | Solves their $\frac{4}{pN+q} = \frac{4}{321}$ to obtain a value for $N$ |
| $(N =) 81$ | A1 | Cao (ignore what they use for $N$) |\begin{enumerate}
\item A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$\begin{aligned}
a _ { 1 } & = 4 \\
a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N }
\end{aligned}$$
(a) Find the values of $a _ { 2 } , a _ { 3 }$ and $a _ { 4 }$
Write your answers as simplified fractions.
Given that
$$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
(b) state the value of $p$ and the value of $q$.\\
(c) Hence calculate the value of $N$ such that $a _ { N } = \frac { 4 } { 321 }$\\
\hfill \mbox{\textit{Edexcel C1 2018 Q6 [7]}}