| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.8 This is a telescoping series question requiring algebraic manipulation to verify an identity, then applying it to find a finite sum and evaluate an infinite series. While the telescoping technique is standard for FP1, the algebraic verification and careful handling of the infinite sum with starting index r=2 require solid technical skill beyond routine A-level work. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks |
|---|---|
| B1, 1 | Obtain given answer correctly |
| Answer | Marks |
|---|---|
| M1 | Express at least 1st two and last term using (i) |
| A1 | All terms correct |
| M1 | Show that correct terms cancel |
| A1, 4 | Obtain correct answer, in terms of \(n\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{4}\) | B1 | Sum to infinity seen or implied |
| B1, 2 | Obtain correct answer |
**Part (i)**
| B1, 1 | Obtain given answer correctly
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**Part (ii)**
| M1 | Express at least 1st two and last term using (i)
| A1 | All terms correct
| M1 | Show that correct terms cancel
| A1, 4 | Obtain correct answer, in terms of $n$
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**Part (iii)**
$\frac{1}{4}$ | B1 | Sum to infinity seen or implied
| B1, 2 | Obtain correct answer
S.C. -3/4 scores B1
---7 (i) Show that $\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.\\
(ii) Hence find an expression, in terms of $n$, for $\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.\\
(iii) Find $\sum _ { r = 2 } ^ { \infty } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.
\hfill \mbox{\textit{OCR FP1 2010 Q7 [7]}}