| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring recognition of telescoping series structure with exponential terms. Part (a) involves algebraic manipulation with powers of 2 and fractions requiring common denominators. Part (b) requires applying the method of differences to sum the series, which is non-trivial as students must recognize the telescoping pattern and carefully track which terms survive. The combination of exponential terms with rational functions makes this harder than standard method of differences questions, but it's still a structured problem with clear guidance. |
| Spec | 4.06b Method of differences: telescoping series |
3
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
\item Hence find
$$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
giving your answer in the form $2 ^ { n } - 1$, where $n$ is an integer.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2012 Q3 [6]}}