| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find parameter from given term |
| Difficulty | Standard +0.3 This is a straightforward recurrence relation problem requiring systematic substitution and algebraic manipulation. Students compute u₂ and u₃ using the given formula, substitute into the constraint equation, and simplify to reach the required quadratic. Part (b) involves solving the quadratic and selecting the appropriate value based on sequence behavior. While it requires careful algebra across multiple steps, the techniques are standard and the path is clear, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u_2 = k - 12\), \(u_3 = k - \dfrac{24}{k-12}\) | M1 | Attempts to apply sequence formula once for either \(u_2\) or \(u_3\) |
| \(u_1 + 2u_2 + u_3 = 0 \Rightarrow 2 + 2(k-12) + k - \dfrac{24}{k-12} = 0\) | dM1 | Attempts to find both \(u_2\) and \(u_3\), then uses \(2 + 2u_2 + u_3 = 0\) to form equation in \(k\) |
| \(\Rightarrow 3k - 22 - \dfrac{24}{k-12} = 0 \Rightarrow (3k-22)(k-12) - 24 = 0\) | A1* | Fully correct work leading to \(3k^2 - 58k + 240 = 0\). Must show at least one correct intermediate line removing fractions. No errors in algebra. |
| \(\Rightarrow 3k^2 - 58k + 240 = 0\) | ||
| (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k = 6,\ \left(\dfrac{40}{3}\right)\) | M1 | Attempts to solve the quadratic; implied by sight of \(k=6\). May use factorisation \((ak\pm c)(bk\pm d)=0\) where \(ab=3, cd=240\), or quadratic formula, or completing the square |
| \(k = 6\) as \(k\) must be an integer | A1 | Chooses \(k=6\) and gives minimal reason e.g. "it is an integer" or "\(\frac{40}{3}\) is not an integer" |
| (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u_3 = 10\) | B1 | Deduces correct value of \(u_3\) |
| (1 mark) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_2 = k - 12$, $u_3 = k - \dfrac{24}{k-12}$ | M1 | Attempts to apply sequence formula once for either $u_2$ or $u_3$ |
| $u_1 + 2u_2 + u_3 = 0 \Rightarrow 2 + 2(k-12) + k - \dfrac{24}{k-12} = 0$ | dM1 | Attempts to find both $u_2$ and $u_3$, then uses $2 + 2u_2 + u_3 = 0$ to form equation in $k$ |
| $\Rightarrow 3k - 22 - \dfrac{24}{k-12} = 0 \Rightarrow (3k-22)(k-12) - 24 = 0$ | A1* | Fully correct work leading to $3k^2 - 58k + 240 = 0$. Must show at least one correct intermediate line removing fractions. No errors in algebra. |
| $\Rightarrow 3k^2 - 58k + 240 = 0$ | | |
| **(3 marks)** | | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k = 6,\ \left(\dfrac{40}{3}\right)$ | M1 | Attempts to solve the quadratic; implied by sight of $k=6$. May use factorisation $(ak\pm c)(bk\pm d)=0$ where $ab=3, cd=240$, or quadratic formula, or completing the square |
| $k = 6$ as $k$ must be an integer | A1 | Chooses $k=6$ and gives minimal reason e.g. "it is an integer" or "$\frac{40}{3}$ is not an integer" |
| **(2 marks)** | | |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_3 = 10$ | B1 | Deduces correct value of $u_3$ |
| **(1 mark)** | | |
---\begin{enumerate}
\item The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$
where $k$ is an integer.\\
Given that $u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0$\\
(a) show that
$$3 k ^ { 2 } - 58 k + 240 = 0$$
(b) Find the value of $k$, giving a reason for your answer.\\
(c) Find the value of $u _ { 3 }$
\hfill \mbox{\textit{Edexcel Paper 1 2021 Q3 [6]}}