| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Moderate -0.8 This is a straightforward arithmetic sequence problem requiring standard formula application (sum of AP) with simple algebraic manipulation. Part (a) uses S_10 = 360 to find a, part (b) substitutes into S_n formula to derive the given quadratic, and part (c) solves it. All steps are routine textbook exercises with no novel insight required, making it easier than average A-level questions. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Uses \(S_{10} = 360 \Rightarrow \frac{10}{2}\{2 \times a + 9 \times 4\} = 360\) | M1 | Uses \(S_{10} = 360\) with correct formula to set up linear equation in \(a\); alternatively uses sum of all 10 terms: \(a+(a+4)+(a+8)+...+(a+36)=360\) |
| \(\Rightarrow 2a + 36 = 72 \Rightarrow a = 18\) | A1 | \(a=18\) following a correct equation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Uses \(S_n = 2146 \Rightarrow \frac{n}{2}\{2 \times \text{"18"} + (n-1) \times 4\} = 2146\) | M1 | Attempts to use formula following through on their value of \(a\) |
| \(\Rightarrow n\{16 + 2n\} = 2146\) | ||
| \(\Rightarrow n^2 + 8n - 1073 = 0\) | A1* | Proceeds to \(n^2+8n-1073=0\) showing sufficient working; one correct simplified intermediate line required |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| States \(29\) | B1 | The \(-37\) if written must be deleted/crossed out, or \(29\) chosen by being ringed or underlined |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Attempts \(\text{"18"} + (\text{"29"}-1) \times 4 = 130\) | M1, A1 | M1: Attempts \(a+(n-1)\times 4\) with their values of \(a\) and \(n\); alternatively uses \(2146 \Rightarrow \frac{29}{2}\{\text{"18"}+l\} \Rightarrow l=...\); A1: Maximum seats \(=130\), may be labelled \(l=130\); scores both marks only if \(a=18\) and \(n=29\) used |
# Question 8:
## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Uses $S_{10} = 360 \Rightarrow \frac{10}{2}\{2 \times a + 9 \times 4\} = 360$ | M1 | Uses $S_{10} = 360$ with correct formula to set up linear equation in $a$; alternatively uses sum of all 10 terms: $a+(a+4)+(a+8)+...+(a+36)=360$ |
| $\Rightarrow 2a + 36 = 72 \Rightarrow a = 18$ | A1 | $a=18$ following a correct equation |
## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Uses $S_n = 2146 \Rightarrow \frac{n}{2}\{2 \times \text{"18"} + (n-1) \times 4\} = 2146$ | M1 | Attempts to use formula following through on their value of $a$ |
| $\Rightarrow n\{16 + 2n\} = 2146$ | | |
| $\Rightarrow n^2 + 8n - 1073 = 0$ | A1* | Proceeds to $n^2+8n-1073=0$ showing sufficient working; one correct simplified intermediate line required |
## Part (c)(i):
| Working | Mark | Guidance |
|---------|------|----------|
| States $29$ | B1 | The $-37$ if written must be deleted/crossed out, or $29$ chosen by being ringed or underlined |
## Part (c)(ii):
| Working | Mark | Guidance |
|---------|------|----------|
| Attempts $\text{"18"} + (\text{"29"}-1) \times 4 = 130$ | M1, A1 | M1: Attempts $a+(n-1)\times 4$ with their values of $a$ and $n$; alternatively uses $2146 \Rightarrow \frac{29}{2}\{\text{"18"}+l\} \Rightarrow l=...$; A1: Maximum seats $=130$, may be labelled $l=130$; scores both marks only if $a=18$ and $n=29$ used |
---\begin{enumerate}
\item In a large theatre there are $n$ rows of seats, where $n$ is a constant.
\end{enumerate}
The number of seats in the first row is $a$, where $a$ is a constant.\\
In each subsequent row there are 4 more seats than in the previous row so that
\begin{itemize}
\item in the 2 nd row there are $( a + 4 )$ seats
\item in the 3rd row there are ( $a + 8$ ) seats
\item the number of seats in each row form an arithmetic sequence
\end{itemize}
Given that the total number of seats in the first 10 rows is 360\\
(a) find the value of $a$.
Given also that the total number of seats in the $n$ rows is 2146\\
(b) show that
$$n ^ { 2 } + 8 n - 1073 = 0$$
(c) Hence\\
(i) state the number of rows of seats in the theatre,\\
(ii) find the maximum number of seats in any one row.
\hfill \mbox{\textit{Edexcel P2 2023 Q8 [7]}}