| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| 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 algebraic manipulation to verify an identity, then applying method of differences to sum a series. While the method of differences is a standard FP2 technique, the algebraic complexity of tracking telescoping terms and simplifying to the given form with unknown constants requires careful bookwork and is more demanding than typical A-level questions. The multi-step nature and need to find constants a and b elevates this above average difficulty. |
| Spec | 4.06b Method of differences: telescoping series |
2. (a) Show that, for $r > 0$
$$r - 3 + \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) }$$
(b) Hence prove, using the method of differences, that
$$\sum _ { r = 1 } ^ { n } \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) } = \frac { n \left( n ^ { 2 } + a n + b \right) } { 2 ( n + 2 ) }$$
where $a$ and $b$ are constants to be found.
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\hfill \mbox{\textit{Edexcel FP2 2016 Q2 [7]}}