| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.8 This is a Further Maths question requiring partial fractions (part i), telescoping series summation (part ii), and solving for N using infinite series convergence (part iii). While the techniques are standard for FP1, the multi-step nature, the need to recognize telescoping behavior, and working with infinite series convergence places it moderately above average difficulty. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks |
|---|---|
| B1 | Show given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Express terms as differences using (i) | |
| M1 | Attempt this for at least first 3 terms | |
| A1 | First 3 terms all correct | |
| A1 | Last 2 terms correct | |
| M1 | Show terms cancelling | |
| \(1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2}\) | A1 | Obtain correct answer, must be in terms of \(n\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{3}{2}\) | B1ft | State or use correct sum to infinity |
| B1 | Their sum to infinity \(-\) their (ii) \(= \frac{11}{30}\) | |
| M1 | Attempt to solve correct equation | |
| \(N = 4\) | A1 | Obtain only \(N = 4\) |
## Question 8:
### Part (i)
| B1 | Show **given** answer correctly
**[1]**
### Part (ii)
| M1 | Express terms as differences using (i)
| M1 | Attempt this for at least first 3 terms
| A1 | First 3 terms all correct
| A1 | Last 2 terms correct
| M1 | Show terms cancelling
$1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2}$ | A1 | Obtain correct answer, must be in terms of $n$
**[6]**
### Part (iii)
$\frac{3}{2}$ | B1ft | State or use correct sum to infinity
| B1 | Their sum to infinity $-$ their (ii) $= \frac{11}{30}$
| M1 | Attempt to solve correct equation
$N = 4$ | A1 | Obtain only $N = 4$
**[4]**
---8 (i) Show that $\frac { 1 } { r } - \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for $\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 2 ) }$.\\
(iii) Given that $\sum _ { r = N + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) } = \frac { 11 } { 30 }$, find the value of $N$.
\hfill \mbox{\textit{OCR FP1 2012 Q8 [11]}}