| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| 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 requiring partial fractions (given in part i), summing to n terms, and finding an infinite tail. While it involves multiple steps and infinite series, the techniques are routine for FP1 students with clear scaffolding provided. Slightly easier than average A-level due to the guided structure. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{1}{r(r+1)}\) | B1 | Show correct process to obtain given result. |
| (ii) \(1 - \frac{1}{n+1}\) | M1 M1 A1 | Express terms as differences using (i). Show all terms cancel. Obtain correct answer, must be \(n\) not any other letter. |
| (iii) \(S_{\infty} = 1\) | B1ft M1 A1 c.a.o. | |
| \(\frac{1}{n+1}\) |
(i) $\frac{1}{r(r+1)}$ | B1 | Show correct process to obtain given result. |
(ii) $1 - \frac{1}{n+1}$ | M1 M1 A1 | Express terms as differences using (i). Show all terms cancel. Obtain correct answer, must be $n$ not any other letter. |
(iii) $S_{\infty} = 1$ | B1ft M1 A1 c.a.o. | | State correct value of sum to infinity. Ft their (ii). Use sum to infinity – their (ii). |
$\frac{1}{n+1}$ | | |
**Total: 3 + 3 + 1 = 7 marks**5 (i) Show that
$$\frac { 1 } { r } - \frac { 1 } { r + 1 } = \frac { 1 } { r ( r + 1 ) }$$
(ii) Hence find an expression, in terms of $n$, for
$$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$
(iii) Hence find the value of $\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }$.
\hfill \mbox{\textit{OCR FP1 2007 Q5 [7]}}