| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Standard summation formulae application |
| Difficulty | Moderate -0.3 This is a straightforward application of standard summation formulae (∑r³, ∑r, ∑r²) that FP1 students are expected to know. Part (a) requires algebraic manipulation to factor the result, and part (b) involves solving a quartic that simplifies nicely. While it requires multiple steps and careful algebra, it's a routine textbook-style question with no novel insight required, making it slightly easier than average. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a) | \(\Sigma r^3 = \Sigma r = \frac{1}{2}n^2(n+1)^2 + \frac{1}{2}n(n+1)\) | M1 |
| Factor \(n\) clearly shown | m1 | or \(n + 1\) clearly shown to be a factor |
| \(... = \frac{1}{4}n(n+1)(n^2 + n + 2)\) | A1A1 | 4 marks |
| 8(b) | Valid equation formed | M1 |
| Factors \(n, n + 1\) removed | m1 | |
| \(3n^2 - 29n - 10 = 0\) | A1 | OE |
| Valid factorisation or solution | m1 | of the correct quadratic |
| \(n = 10\) is the only pos int solution | A1 | 5 marks |
| Total for Q8 | 9 marks |
8(a) | $\Sigma r^3 = \Sigma r = \frac{1}{2}n^2(n+1)^2 + \frac{1}{2}n(n+1)$ | M1 | at least one term correct |
| Factor $n$ clearly shown | m1 | or $n + 1$ clearly shown to be a factor |
| $... = \frac{1}{4}n(n+1)(n^2 + n + 2)$ | A1A1 | 4 marks | OE; A1 for $\frac{1}{4}$, A1 for quadratic |
8(b) | Valid equation formed | M1 |
| Factors $n, n + 1$ removed | m1 | |
| $3n^2 - 29n - 10 = 0$ | A1 | OE |
| Valid factorisation or solution | m1 | of the correct quadratic |
| $n = 10$ is the only pos int solution | A1 | 5 marks | SC 1/2 for $n = 10$ after correct quad |
| **Total for Q8** | **9 marks** |
---8
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$
can be expressed in the form
$$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$
where $k$ is a rational number and $a , b$ and $c$ are integers.
\item Show that there is exactly one positive integer $n$ for which
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q8 [9]}}