| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.3 This is a straightforward method of differences question with the partial fraction decomposition already provided. Part (i) requires only algebraic manipulation to verify the given identity, and part (ii) is a standard telescoping series application where most terms cancel. The mechanics are routine for Further Maths students who have learned this technique. |
| Spec | 1.02y Partial fractions: decompose rational functions4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2r+1} - \frac{1}{2r+3} = \frac{2r+3-(2r+1)}{(2r+1)(2r+3)} = \frac{2}{(2r+1)(2r+3)}\) | M1, A1 | Attempt at common denominator |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum_{r=1}^{30} \frac{1}{(2r+1)(2r+3)} = \frac{1}{2}\sum_{r=1}^{30}\left[\frac{1}{2r+1} - \frac{1}{2r+3}\right]\) | M1 | Use of (i); do not penalise missing factor of \(\frac{1}{2}\) |
| \(= \frac{1}{2}\left[\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\cdots+\left(\frac{1}{59}-\frac{1}{61}\right)+\left(\frac{1}{61}-\frac{1}{63}\right)\right]\) | M1 | Sufficient terms to show pattern |
| \(= \frac{1}{2}\left(\frac{1}{3}-\frac{1}{63}\right) = \frac{10}{63}\) | M1, A1, A1 | Cancelling terms; Factor \(\frac{1}{2}\) used; oe cao |
# Question 5:
## Part (i):
$\frac{1}{2r+1} - \frac{1}{2r+3} = \frac{2r+3-(2r+1)}{(2r+1)(2r+3)} = \frac{2}{(2r+1)(2r+3)}$ | M1, A1 | Attempt at common denominator
**[2]**
## Part (ii):
$\sum_{r=1}^{30} \frac{1}{(2r+1)(2r+3)} = \frac{1}{2}\sum_{r=1}^{30}\left[\frac{1}{2r+1} - \frac{1}{2r+3}\right]$ | M1 | Use of (i); do not penalise missing factor of $\frac{1}{2}$
$= \frac{1}{2}\left[\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\cdots+\left(\frac{1}{59}-\frac{1}{61}\right)+\left(\frac{1}{61}-\frac{1}{63}\right)\right]$ | M1 | Sufficient terms to show pattern
$= \frac{1}{2}\left(\frac{1}{3}-\frac{1}{63}\right) = \frac{10}{63}$ | M1, A1, A1 | Cancelling terms; Factor $\frac{1}{2}$ used; oe cao
**[5]**
---5\\
(i) Show that $\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }$.\\
(ii) Use the method of differences to find $\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$, expressing your answer as a fraction.
\hfill \mbox{\textit{OCR MEI FP1 2012 Q5 [7]}}