| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Standard summation formulae application |
| Difficulty | Moderate -0.5 This is a routine Further Maths question using standard summation techniques. Part (a)(i) is trivial algebra, part (a)(ii) requires applying standard Σr and Σr² formulae (core FP1 content), and part (b) is straightforward substitution. While it's multi-step, it follows a completely standard template with no problem-solving insight required, making it easier than average even for Further Maths. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
| Answer | Marks | Guidance |
|---|---|---|
| \((2r - 1)^2 = 4r^2 - 4r + 1\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum (2r - 1)^2 = 4\sum r^2 - 4\sum r + \sum 1\) | M1 | |
| \(\ldots = \frac{4}{3}n^3 - \frac{4}{3}n + \sum 1\) | m1A1 | |
| \(\sum 1 = n\) | B1 | |
| Result convincingly shown | A1 | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Sum \(= f(100) - f(50)\) | M1A1 | |
| \(\ldots = 1166 650\) | A2 | 4 marks |
### Part (a)(i)
$(2r - 1)^2 = 4r^2 - 4r + 1$ | B1 | 1 mark |
### Part (a)(ii)
$\sum (2r - 1)^2 = 4\sum r^2 - 4\sum r + \sum 1$ | M1 | |
$\ldots = \frac{4}{3}n^3 - \frac{4}{3}n + \sum 1$ | m1A1 | |
$\sum 1 = n$ | B1 | |
Result convincingly shown | A1 | 5 marks | AG
### Part (b)
Sum $= f(100) - f(50)$ | M1A1 | |
$\ldots = 1166 650$ | A2 | 4 marks | SC f(100) - f(51) = 1 156 449; 3/4
### **Total for Question 6: 10 marks**
---6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Expand $( 2 r - 1 ) ^ { 2 }$.
\item Hence show that
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
\end{enumerate}\item Hence find the sum of the squares of the odd numbers between 100 and 200 .
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q6 [10]}}