CAIE FP1 2013 November — Question 1 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPartial Fractions
TypeThree linear factors in denominator
DifficultyStandard +0.8 This is a Further Maths question requiring partial fractions decomposition with three linear factors, followed by telescoping series summation. While the partial fractions setup is standard, recognizing and executing the telescoping pattern requires more sophistication than typical A-level questions, and the infinite series conclusion adds another layer. This is moderately challenging for Further Maths but not exceptionally difficult.
Spec1.02y Partial fractions: decompose rational functions4.06b Method of differences: telescoping series
1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$
Question 1:
AnswerMarks Guidance
Working/AnswerMarks Guidance
\(\frac{1}{r(r-1)(r+1)} = \frac{1}{2(r-1)} - \frac{1}{r} + \frac{1}{2(r+1)}\)B1 (1)
\(\left(\frac{1}{2}-\frac{1}{2}+\frac{1}{6}\right)+\left(\frac{1}{4}-\frac{1}{3}+\frac{1}{8}\right)\cdots\left(\frac{1}{2(n-1)}-\frac{1}{n}+\frac{1}{2(n+1)}\right)\)M1A1 Expresses each term in fractions
\(= \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n+1)}\)M1A1 (4) OE
\(S_\infty = \frac{1}{4}\)B1 (1)
Total: [6]
## Question 1:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\frac{1}{r(r-1)(r+1)} = \frac{1}{2(r-1)} - \frac{1}{r} + \frac{1}{2(r+1)}$ | B1 | **(1)** |
| $\left(\frac{1}{2}-\frac{1}{2}+\frac{1}{6}\right)+\left(\frac{1}{4}-\frac{1}{3}+\frac{1}{8}\right)\cdots\left(\frac{1}{2(n-1)}-\frac{1}{n}+\frac{1}{2(n+1)}\right)$ | M1A1 | Expresses each term in fractions |
| $= \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n+1)}$ | M1A1 | **(4)** OE |
| $S_\infty = \frac{1}{4}$ | B1 | **(1)** |

**Total: [6]**

---
1 Express $\frac { 1 } { r ( r + 1 ) ( r - 1 ) }$ in partial fractions.

Find

$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

State the value of

$$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

\hfill \mbox{\textit{CAIE FP1 2013 Q1 [6]}}
This paper (12 questions)
View full paper
Q1 6 Q2 6 Q3 7 Q4 7 Q5 8 Q6 8 Q7 9 Q8 11 Q9 11 Q10 13 Q11 EITHER Q11 OR