| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Simple recurrence evaluation |
| Difficulty | Moderate -0.3 Part (a) requires straightforward substitution into a recurrence relation (3 iterations), and part (b) requires recognizing the sequence alternates between two values (3 and -2), making the sum a simple arithmetic calculation. This is easier than average as it involves only direct computation and pattern recognition, with no proof or novel problem-solving required. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempts iteration formula at least once: \(u_{1+1}=2-\frac{4}{3}=\ldots\) | M1 | Implied by correct value on correct term, or correct follow-through value |
| Any one correct value on correct term: \(u_2=\frac{2}{3}\) (condone 0.67) or \(u_3=-4\) or \(u_4=3\) | A1 | Watch for correct values on wrong terms |
| All three correct and labelled: \(u_2=\frac{2}{3}\), \(u_3=-4\), \(u_4=3\) | A1 | May be awarded from part (b) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\displaystyle\sum_{r=1}^{100}u_r = 33\times\left(3+\frac{2}{3}+(-4)\right)+3\) | M1 | Correct method to find sum of 100 terms; listing acceptable only if all terms present or correct result given |
| \(= -8\) | A1 | \(-8\) alone scores both marks following correct (a), provided no incorrect method |
# Question 2:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempts iteration formula at least once: $u_{1+1}=2-\frac{4}{3}=\ldots$ | M1 | Implied by correct value on correct term, or correct follow-through value |
| Any one correct value on correct term: $u_2=\frac{2}{3}$ (condone 0.67) or $u_3=-4$ or $u_4=3$ | A1 | Watch for correct values on wrong terms |
| All three correct and labelled: $u_2=\frac{2}{3}$, $u_3=-4$, $u_4=3$ | A1 | May be awarded from part (b) |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\displaystyle\sum_{r=1}^{100}u_r = 33\times\left(3+\frac{2}{3}+(-4)\right)+3$ | M1 | Correct method to find sum of 100 terms; listing acceptable only if all terms present or correct result given |
| $= -8$ | A1 | $-8$ alone scores both marks following correct (a), provided no incorrect method |
---\begin{enumerate}
\item A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$\begin{gathered}
u _ { 1 } = 3 \\
u _ { n + 1 } = 2 - \frac { 4 } { u _ { n } }
\end{gathered}$$
(a) Find the value of $u _ { 2 }$, the value of $u _ { 3 }$ and the value of $u _ { 4 }$\\
(b) Find the value of
$$\sum _ { r = 1 } ^ { 100 } u _ { r }$$
\hfill \mbox{\textit{Edexcel P2 2023 Q2 [5]}}