| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| 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 standard telescoping series question requiring algebraic manipulation to verify an identity, then applying it to find a sum. Part (i) is routine algebra, part (ii) is a textbook telescoping sum application, and part (iii) is immediate once (ii) is done. While it's FP1, this is a straightforward example of a well-known technique with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{(r+1)^2 - r(r+2)}{(r+2)(r+1)}\) | M1 | Show correct process for subtracting fractions |
| \(\frac{1}{(r+1)(r+2)}\) | A1 | Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2}{3} - \frac{1}{2} + \frac{3}{4} - \frac{2}{3} + \ldots + \frac{n+1}{n+2} - \frac{n}{n+1}\) | M1 | Express terms as differences using (i) |
| A1 | At least first two and last term correct | |
| \(\frac{n+1}{n+2} - \frac{1}{2}\) | M1 | Show or imply that pairs of terms cancel |
| A1 | Obtain correct answer in any form |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum_{r=1}^{n} u_r = f(n+1) - f(1)\) | M2 | State that \(\sum_{r=1}^{n} u_r = f(n+1) - f(1)\) |
| A1A1 | Each term correct | |
| B1 ft | Obtain value from their sum to \(n\) terms |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2}\) | A1 | Obtain correct answer in any form |
### (i)
$\frac{(r+1)^2 - r(r+2)}{(r+2)(r+1)}$ | M1 | Show correct process for subtracting fractions
$\frac{1}{(r+1)(r+2)}$ | A1 | Obtain given answer correctly
**Subtotal: 2 marks**
### (ii)
EITHER:
$\frac{2}{3} - \frac{1}{2} + \frac{3}{4} - \frac{2}{3} + \ldots + \frac{n+1}{n+2} - \frac{n}{n+1}$ | M1 | Express terms as differences using (i)
| A1 | At least first two and last term correct
$\frac{n+1}{n+2} - \frac{1}{2}$ | M1 | Show or imply that pairs of terms cancel
| A1 | Obtain correct answer in any form
**Subtotal: 4 marks**
OR:
$\sum_{r=1}^{n} u_r = f(n+1) - f(1)$ | M2 | State that $\sum_{r=1}^{n} u_r = f(n+1) - f(1)$
| A1A1 | Each term correct
| B1 ft | Obtain value from their sum to $n$ terms
**Subtotal: 4 marks**
### (iii)
$\frac{1}{2}$ | A1 | Obtain correct answer in any form
**Subtotal: 1 mark**
**Total: 7 marks**\begin{enumerate}[label=(\roman*)]
\item Show that
$$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$
[2]
\item Hence find an expression, in terms of $n$, for
$$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$
[4]
\item Hence write down the value of $\sum_{r=1}^{\infty} \frac{1}{(r+1)(r+2)}$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2005 Q5 [7]}}