| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.8 This is a standard FP1 method of differences question with three parts: algebraic verification (routine), telescoping sum proof (requires careful bookkeeping of terms but follows standard technique), and infinite series evaluation (straightforward application). While it requires multiple steps and careful algebra, it follows a well-established pattern taught in FP1 with no novel insight needed. Slightly above average difficulty due to the algebraic manipulation and tracking of telescoping terms. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks |
|---|---|
| M1, A1 [2] | Obtain correct numerator from addition or partial fractions; Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\frac{n}{(n+1)(n+2)}\) | M1, A1, A1; M1, A1 [5] | Express at least three relevant terms using (i); 1st three terms correct; Last two terms correct; Show correct cancelling; Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\frac{1}{6}\) | M1, A1 [2] | Sum 1 to ∞ - 1st term or start process at \(r=2\); Obtain correct answer |
### (i)
| M1, A1 [2] | Obtain correct numerator from addition or partial fractions; Obtain given answer correctly
### (ii)
Answer: $\frac{n}{(n+1)(n+2)}$ | M1, A1, A1; M1, A1 [5] | Express at least three relevant terms using (i); 1st three terms correct; Last two terms correct; Show correct cancelling; Obtain given answer correctly
### (iii)
Answer: $\frac{1}{6}$ | M1, A1 [2] | Sum 1 to ∞ - 1st term or start process at $r=2$; Obtain correct answer\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{1}{r} - \frac{3}{r+1} + \frac{2}{r+2} = \frac{2-r}{r(r+1)(r+2)}$. [2]
\item Hence show that $\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = -\frac{n}{(n+1)(n+2)}$. [5]
\item Find the value of $\sum_{r=3}^{\infty} \frac{2-r}{r(r+1)(r+2)}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q8 [9]}}