| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.3 This is a straightforward method of differences question where the partial fraction form is already given. Part (a) requires simple algebraic manipulation to find constants A and B, and part (b) is a standard telescoping sum application. While it's a Further Maths topic, the execution is mechanical with no novel insight required, making it slightly easier than average. |
| Spec | 4.05c Partial fractions: extended to quadratic denominators4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(r^2 + r - 1 = A(r^2 + r) + B\) with \(A = 1, B = -1\) | M1, A1, A1F | Any correct method; ft B if incorrect A and vice versa. Or \(\frac{r^2 + r - 1}{r^2 + r} = 1 - \frac{1}{r(r+1)} = 1 - \left(\frac{1}{r} - \frac{1}{r+1}\right)\) |
| Answer | Marks | Guidance |
|---|---|---|
| - Sum \(= 98 + \frac{1}{100} = 98.01\) | M1, A1F, m1, A1F | Do not allow M1 if merely \(\sum\frac{1}{r} - \sum\frac{1}{r+1}\) is summed. A1 for suitable (3 at least) number of rows. Must have 98 or 99. OE Allow correct answer with no working 4 marks |
**(a)** $r^2 + r - 1 = A(r^2 + r) + B$ with $A = 1, B = -1$ | M1, A1, A1F | Any correct method; ft B if incorrect A and vice versa. Or $\frac{r^2 + r - 1}{r^2 + r} = 1 - \frac{1}{r(r+1)} = 1 - \left(\frac{1}{r} - \frac{1}{r+1}\right)$ | B1, M1A1
**(b)**
- $r = 1$: $1 - \frac{1}{1} + \frac{1}{2}$
- $r = 2$: $1 - \frac{1}{2} + \frac{1}{3}$
- $r = 99$: $1 - \frac{1}{99} + \frac{1}{100}$
- Sum $= 98 + \frac{1}{100} = 98.01$ | M1, A1F, m1, A1F | Do not allow M1 if merely $\sum\frac{1}{r} - \sum\frac{1}{r+1}$ is summed. A1 for suitable (3 at least) number of rows. Must have 98 or 99. OE Allow correct answer with no working 4 marks
**Total: 7 marks**
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\begin{enumerate}[label=(\alph*)]
\item Given that
$$\frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) } = A + B \left( \frac { 1 } { r } - \frac { 1 } { r + 1 } \right)$$
find the values of $A$ and $B$.
\item Hence find the value of
$$\sum _ { r = 1 } ^ { 99 } \frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q1 [7]}}