| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Challenging +1.3 This is a standard FP1 telescoping series question with three routine parts: algebraic verification, finding a finite sum using telescoping, and evaluating an infinite series. While it requires careful bookkeeping and understanding of partial fractions in reverse, the method is well-practiced and follows a predictable pattern. Slightly above average difficulty due to the multi-step nature and being Further Maths content, but still a textbook exercise. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| M1 | Use correct common denominator, numerator must be quadratic | |
| A1 | Obtain given result | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| 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 | |
| \(\frac{7}{2} - \frac{3}{n} - \frac{1}{n+1}\) | M1 | Show terms cancelling; Need not be tidied up |
| A1 | Obtain correct answer, must be in terms of \(n\) | |
| [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{5}{4}\) | M1 | Attempt to start summation at correct term; Could be \(\sum_2^\infty - \sum_2^3\) |
| A1 | Obtain correct answer from correct working | |
| [2] |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Use correct common denominator, numerator must be quadratic |
| | A1 | Obtain **given** result |
| **[2]** | | |
---
## Question 8(ii):
| 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 |
| $\frac{7}{2} - \frac{3}{n} - \frac{1}{n+1}$ | M1 | Show terms cancelling; Need not be tidied up |
| | A1 | Obtain correct answer, must be in terms of $n$ |
| **[6]** | | |
---
## Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{5}{4}$ | M1 | Attempt to start summation at correct term; Could be $\sum_2^\infty - \sum_2^3$ |
| | A1 | Obtain correct answer from correct working |
| **[2]** | | |
---8 (i) Show that $\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.\\
(ii) Hence find an expression, in terms of $n$, for $\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.\\
(iii) Hence find the value of $\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.
\hfill \mbox{\textit{OCR FP1 2015 Q8 [10]}}