| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Find term or common difference |
| Difficulty | Moderate -0.8 This is a straightforward arithmetic series question requiring only standard formula application (S_n = n/2(2a + (n-1)d)) and solving simultaneous equations. Part (a) is essentially given, and part (b) involves writing one more equation (a+d + a+6d = 13) then solving a simple 2×2 system. No problem-solving insight required, just routine algebraic manipulation of well-known formulas. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | \(\{S_{29} = \}\frac{29}{2}[2a + 28d]\) | M1 |
| \(29(a + 14d) = 1102\) | m1 | |
| \(a + 14d = \frac{1102}{29} \Rightarrow a + 14d = 38\) | A1 | 3 marks |
| 4(b) | \(u_2 = a + d\) or \(u_2 = a + 6d\) | B1 |
| \(u_2 + u_7 = 13 \Rightarrow 2a + 7d = 13\) | M1 | |
| e.g. \(21d = 63; \quad 3a = -12\) | m1 | |
| \(a = -4 \quad d = 3\) | A1 | 4 marks |
**4(a)** | $\{S_{29} = \}\frac{29}{2}[2a + 28d]$ | M1 | | Formula for $S_n$ with $n = 29$ substituted and with $a$ and $d$ |
| | $29(a + 14d) = 1102$ | m1 | | Equation formed then some manipulation |
| | $a + 14d = \frac{1102}{29} \Rightarrow a + 14d = 38$ | A1 | 3 marks | CSO AG |
**4(b)** | $u_2 = a + d$ or $u_2 = a + 6d$ | B1 | | Either expression correct |
| | $u_2 + u_7 = 13 \Rightarrow 2a + 7d = 13$ | M1 | | Forming equation using $u_2$ & $u_7$ both in form $a + kd$ |
| | e.g. $21d = 63; \quad 3a = -12$ | m1 | | Solving $a + 14d = 38$ with candidate's '$2a + 7d = 13'$ to at least stage of elimination of either $a$ or $d$ |
| | $a = -4 \quad d = 3$ | A1 | 4 marks | Both correct |
---4 An arithmetic series has first term $a$ and common difference $d$.\\
The sum of the first 29 terms is 1102.
\begin{enumerate}[label=(\alph*)]
\item Show that $a + 14 d = 38$.
\item The sum of the second term and the seventh term is 13 .
Find the value of $a$ and the value of $d$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q4 [7]}}