| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Proving standard summation formulae |
| Difficulty | Standard +0.3 This is a standard Further Maths telescoping series proof with clear scaffolding. Part (a) is routine algebra, part (b) follows a well-established method once the telescoping pattern is recognized, and part (c) is straightforward application. While it requires more sophistication than basic A-level, the technique is commonly taught and practiced in FP2. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^34.06b Method of differences: telescoping series |
2. Given that for all real values of $r , \quad ( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = A r ^ { 2 } + B$, where $A$ and $B$ are constants,
\begin{enumerate}[label=(\alph*)]
\item find the value of $A$ and the value of $B$.
\item Hence, or otherwise, prove that $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$.
\item Calculate $\sum _ { r = 1 } ^ { 40 } ( 3 r - 1 ) ^ { 2 }$.\\
(3)(Total 10 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2006 Q2 [10]}}