| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Challenging +1.2 This is a standard method of differences question with the identity provided. Part (a) is routine algebraic verification (expand and simplify), and part (b) applies the telescoping sum technique directly. While it requires careful algebraic manipulation and understanding of the method, it's a textbook application of a Further Maths technique with no novel insight needed. |
| Spec | 4.06b Method of differences: telescoping series |
4
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { f } ( r ) = r ^ { 2 } \left( 2 r ^ { 2 } - 1 \right)$, show that
$$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = ( 2 r - 1 ) ^ { 3 }$$
\item Use the method of differences to show that
$$\sum _ { r = n + 1 } ^ { 2 n } ( 2 r - 1 ) ^ { 3 } = 3 n ^ { 2 } \left( 10 n ^ { 2 } - 1 \right)$$
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q4 [7]}}