| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Surd rationalization method of differences |
| Difficulty | Standard +0.3 This is a standard FP1 telescoping series question with routine algebraic manipulation. Part (i) requires straightforward rationalisation of the denominator (a technique students know well). Part (ii) is a textbook telescoping sum where most terms cancel, leaving only boundary terms. Part (iii) is immediate once part (ii) is done. While it's Further Maths content, this is a very standard exercise type with no novel insight required—slightly easier than average overall. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks |
|---|---|
| M1 | Attempt to rationalise denominator or cross multiply |
| A1 2 | Obtain given answer correctly |
| Answer | Marks |
|---|---|
| M1 | Express terms as differences using (i) |
| M1 | Attempt this for at least 1st three terms |
| A1 | 1st three terms all correct |
| A1 | Last two terms all correct |
| M1 | Show pairs cancelling |
| A1 6 | Obtain correct answer, in terms of \(n\) |
| Answer | Marks |
|---|---|
| B1 1 | Sensible statement for divergence |
## (i)
| M1 | Attempt to rationalise denominator or cross multiply
| A1 2 | Obtain given answer correctly
## (ii)
| M1 | Express terms as differences using (i)
| M1 | Attempt this for at least 1st three terms
| A1 | 1st three terms all correct
| A1 | Last two terms all correct
| M1 | Show pairs cancelling
| A1 6 | Obtain correct answer, in terms of $n$
$\frac{1}{2}(\sqrt{n+2} + \sqrt{n+1} - \sqrt{2} - 1)$
## (iii)
| B1 1 | Sensible statement for divergence
---\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{1}{\sqrt{r + 2} + \sqrt{r}} = \frac{\sqrt{r + 2} - \sqrt{r}}{2}$. [2]
\item Hence find an expression, in terms of $n$, for
$$\sum_{r=1}^{n} \frac{1}{\sqrt{r + 2} + \sqrt{r}}.$$ [6]
\item State, giving a brief reason, whether the series $\sum_{r=1}^{\infty} \frac{1}{\sqrt{r + 2} + \sqrt{r}}$ converges. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2010 Q8 [9]}}