| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| 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 routine algebraic verification, part (ii) applies the telescoping technique to find a finite sum (a common FP1 exercise), and part (iii) uses the limit of the finite sum to find the infinite tail. While it requires understanding of partial fractions and series convergence, the question structure guides students through each step without requiring novel insight. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Combine with a common denominator | M1 | |
| Obtain given answer correctly | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{n}{n+1}\) | M1, A1, M1, A1 | Express terms using (i); At least 1st two and last two correct; Show terms cancelling; Obtain correct answer in terms of \(n\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - \dfrac{n}{n+1}\) | B1 | \(\lim_{n\to\infty}\frac{n}{n+1}=1\) |
| B1FT | This value \(-\) (ii) | |
| [2] |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Combine with a common denominator | M1 | |
| Obtain given answer correctly | A1 | |
| **[2]** | | |
---
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{n}{n+1}$ | M1, A1, M1, A1 | Express terms using (i); At least 1st two and last two correct; Show terms cancelling; Obtain correct answer in terms of $n$ |
| **[4]** | | |
---
## Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - \dfrac{n}{n+1}$ | B1 | $\lim_{n\to\infty}\frac{n}{n+1}=1$ |
| | B1FT | This value $-$ (ii) |
| **[2]** | | |
---8 (i) Show that $\frac { r } { r + 1 } - \frac { r - 1 } { r } \equiv \frac { 1 } { r ( r + 1 ) }$.\\
(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 $\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }$.\\
\hfill \mbox{\textit{OCR FP1 2012 Q8 [8]}}