| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sequence defined by formula |
| Difficulty | Easy -1.2 This is a straightforward arithmetic sequence question requiring only direct substitution to find terms, identification of the common difference from the formula, and solving a standard sum equation. All parts are routine applications of basic formulas with no problem-solving insight needed, making it easier than average but not trivial since part (c) requires setting up and solving a quadratic equation. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | \(u_1 = 87; u_2 = 84\) | B1;B1 ft on \(u_2 = u_1 - 3\) |
| 3(b) | Common difference (d) is −3 | B1 |
| 3(c) | \(\sum_{n=1}^{k} u_n\) = sum of AP; \(........= \frac{k}{2}[174 + (k-1)(-3)]\); 0 = \(\frac{k}{2}[177 - 3k]\) ⟹ 177 = 3k ⟹ k = 59 | M1, A1 ft, A1 |
| ALT1 | \(= \sum_{n=1}^{k} 90 - \sum_{n=1}^{k} 3n = 90k - 3[\frac{k}{2}(k+1)]\); 0 = 90k − 1.5k(k+1) ⟹ k = 59 | M1;A1, A1 |
3(a) | $u_1 = 87; u_2 = 84$ | B1;B1 ft on $u_2 = u_1 - 3$ | SC B1 for 90, 87 |
3(b) | Common difference (d) is −3 | B1 | |
3(c) | $\sum_{n=1}^{k} u_n$ = sum of AP; $........= \frac{k}{2}[174 + (k-1)(-3)]$; 0 = $\frac{k}{2}[177 - 3k]$ ⟹ 177 = 3k ⟹ k = 59 | M1, A1 ft, A1 | OE ft on $u_1$ and use of d = 3 (For M1A1 ft condone n in place of k); Just the single value 59 |
ALT1 | $= \sum_{n=1}^{k} 90 - \sum_{n=1}^{k} 3n = 90k - 3[\frac{k}{2}(k+1)]$; 0 = 90k − 1.5k(k+1) ⟹ k = 59 | M1;A1, A1 | M1 split and either 90k or $[\frac{k}{2}(k+1)]$; (For 1st two marks condone n in place of k) |3 The $n$th term of an arithmetic sequence is $u _ { n }$, where
$$u _ { n } = 90 - 3 n$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $u _ { 1 }$ and the value of $u _ { 2 }$.
\item Write down the common difference of the arithmetic sequence.
\item Given that $\sum _ { n = 1 } ^ { k } u _ { n } = 0$, find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2005 Q3 [6]}}