| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Sum from n+1 to 2n or similar range |
| Difficulty | Standard +0.3 This is a straightforward application of standard summation formulae. Part (a) requires algebraic manipulation to derive one formula from two given ones (expanding r²(r-1) = r³ - r²), and part (b) applies the difference of sums technique (sum from 1 to 11 minus sum from 1 to 3). Both are routine techniques for FP1 with no novel insight required, making this slightly easier than average. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma r(r-1) = \Sigma r^3 - \Sigma r^2\) Good progress with expansion. Factors \(n\) and \(n+1\) found; \(\ldots = \frac{1}{12}n(n^2-1)(3n+2)\) | M1, m1, A1, A1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(f(11) - f(3)\) in above; \(f(11) = 3850\); \(f(3) = 22\) (so answer is 3828) | M1, A1, A1 | 3 marks |
**Part (a)**
| $\Sigma r(r-1) = \Sigma r^3 - \Sigma r^2$ Good progress with expansion. Factors $n$ and $n+1$ found; $\ldots = \frac{1}{12}n(n^2-1)(3n+2)$ | M1, m1, A1, A1 | 4 marks | With attempt to use given formulae or use of common factors; Allow verification here; Convincingly shown (AG) |
**Part (b)**
| Use of $f(11) - f(3)$ in above; $f(11) = 3850$; $f(3) = 22$ (so answer is 3828) | M1, A1, A1 | 3 marks | M1 for $f(11) - f(4)$; PI by correct answer; ditto |
**Total: 7 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item Use the formulae
$$\begin{gathered}
\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) \\
\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }
\end{gathered}$$
and
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$
(4 marks)
\item Use the result from part (a) to find the value of
$$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q3 [7]}}