| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Show quadratic equation in n |
| Difficulty | Moderate -0.3 This is a straightforward multi-part arithmetic series question requiring standard formula application (nth term and sum formulas) with routine algebraic manipulation. Part (c) involves showing a given quadratic equation rather than deriving it independently, and part (d) is simple factorization. Slightly easier than average due to the scaffolded structure and 'show that' format reducing problem-solving demand. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a + 17d = 25\) and \(a + 20d = 32.5\) | B1, B1 | (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3d = 7.5\), so \(\underline{d = 2.5}\) | M1 | Full method leading to solution for \(d\) or \(a\) without assuming \(a = -17.5\) |
| \(a = 32.5 - 20 \times 2.5\), so \(\underline{a = -17.5}\) | A1cso | Finding correct values for both \(a\) and \(d\) with no incorrect working (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2750 = \frac{n}{2}\left[-35 + \frac{5}{2}(n-1)\right]\) | M1, A1ft | For attempt to form equation with correct \(S_n\) formula and 2750, with values of \(a\) and \(d\) |
| \(4 \times 2750 = n(5n - 75)\); \(4 \times 550 = n(n-15)\) | M1 | For expanding and simplifying to 3-term quadratic |
| \(n^2 - 15n = 55 \times 40\) | A1cso | (4 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n^2 - 15n - 55 \times 40 = 0\) or \(n^2 - 15n - 2200 = 0\) | M1 | Forming correct 3TQ \(= 0\) |
| \((n-55)(n+40) = 0\) | M1 | For attempt to solve 3TQ |
| \(\underline{n = 55}\) (ignore \(-40\)) | A1 | \(n = 55\) dependent on both M marks; no working or trial and improvement scoring 55 scores all 3 marks (3 marks total) |
# Question 9:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a + 17d = 25$ and $a + 20d = 32.5$ | B1, B1 | (2 marks total) |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3d = 7.5$, so $\underline{d = 2.5}$ | M1 | Full method leading to solution for $d$ or $a$ without assuming $a = -17.5$ |
| $a = 32.5 - 20 \times 2.5$, so $\underline{a = -17.5}$ | A1cso | Finding correct values for both $a$ and $d$ with no incorrect working (2 marks total) |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2750 = \frac{n}{2}\left[-35 + \frac{5}{2}(n-1)\right]$ | M1, A1ft | For attempt to form equation with correct $S_n$ formula and 2750, with values of $a$ and $d$ |
| $4 \times 2750 = n(5n - 75)$; $4 \times 550 = n(n-15)$ | M1 | For expanding and simplifying to 3-term quadratic |
| $n^2 - 15n = 55 \times 40$ | A1cso | (4 marks total) |
## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n^2 - 15n - 55 \times 40 = 0$ or $n^2 - 15n - 2200 = 0$ | M1 | Forming correct 3TQ $= 0$ |
| $(n-55)(n+40) = 0$ | M1 | For attempt to solve 3TQ |
| $\underline{n = 55}$ (ignore $-40$) | A1 | $n = 55$ dependent on both M marks; no working or trial and improvement scoring 55 scores all 3 marks (3 marks total) |
---9. The first term of an arithmetic series is $a$ and the common difference is $d$.
The 18th term of the series is 25 and the 21st term of the series is $32 \frac { 1 } { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Use this information to write down two equations for $a$ and $d$.
\item Show that $a = - 17.5$ and find the value of $d$.
The sum of the first $n$ terms of the series is 2750 .
\item Show that $n$ is given by
$$n ^ { 2 } - 15 n = 55 \times 40 .$$
\item Hence find the value of $n$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2009 Q9 [11]}}