| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.3 This is a standard telescoping series question with clear scaffolding. Part (i) is trivial algebraic verification, part (ii) is a routine telescoping sum application (a core FP1 technique), and part (iii) simply requires taking the limit as n→∞ and basic manipulation. The question guides students through each step with no novel insight required—slightly easier than average for FP1. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{r+2-r}{r(r+2)} = \frac{2}{r(r+2)}\) | M1 A1 | Show correct process for subtracting fractions; Obtain given answer correctly |
| (ii) (Express terms as differences using (i)) | M1 | Express terms as differences using (i) |
| (Express 1st 3 (or last 3) terms so that cancelling occurs) | M1 | Express 1st 3 (or last 3) terms so that cancelling occurs |
| \(1 + \frac{1}{2}\) | A1 | Obtain \(1 + \frac{1}{2}\) |
| \(-\frac{1}{n+2}, -\frac{1}{n+1}\) | A1 | Obtain \(-\frac{1}{n+2}, -\frac{1}{n+1}\) |
| \(\frac{3}{2} - \frac{1}{n+1} - \frac{1}{n+2}\) | A1 | Obtain correct answer in any form |
| (iii) (a) \(\frac{3}{2}\) | B1ft | Obtain value from their sum to n terms |
| (b) \(\frac{1}{n+1} + \frac{1}{n+2}\) | M1 A1ft | Using (iii)(a) – (ii) or method of differences again [\(n \to \infty\) is a method error]; Obtain answer in any form |
(i) $\frac{r+2-r}{r(r+2)} = \frac{2}{r(r+2)}$ | M1 A1 | Show correct process for subtracting fractions; Obtain given answer correctly
(ii) (Express terms as differences using (i)) | M1 | Express terms as differences using (i)
(Express 1st 3 (or last 3) terms so that cancelling occurs) | M1 | Express 1st 3 (or last 3) terms so that cancelling occurs
$1 + \frac{1}{2}$ | A1 | Obtain $1 + \frac{1}{2}$
$-\frac{1}{n+2}, -\frac{1}{n+1}$ | A1 | Obtain $-\frac{1}{n+2}, -\frac{1}{n+1}$
$\frac{3}{2} - \frac{1}{n+1} - \frac{1}{n+2}$ | A1 | Obtain correct answer in any form
(iii) (a) $\frac{3}{2}$ | B1ft | Obtain value from their sum to n terms
(b) $\frac{1}{n+1} + \frac{1}{n+2}$ | M1 A1ft | Using (iii)(a) – (ii) or method of differences again [$n \to \infty$ is a method error]; Obtain answer in any form
**Total: 10 marks**9 (i) Show that $\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for
$$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$
(iii) Hence find the value of
\begin{enumerate}[label=(\alph*)]
\item $\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$,
\item $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2006 Q9 [10]}}