CAIE FP1 2016 June — Question 2 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2016
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPartial Fractions
TypeThree linear factors in denominator
DifficultyStandard +0.3 This is a standard Further Maths partial fractions question with three linear factors followed by telescoping series summation. The partial fractions decomposition is routine, and recognizing the telescoping pattern is a well-practiced technique at this level. The infinite sum follows immediately from the finite case. Slightly above average difficulty due to being Further Maths content, but this is a textbook example of the method.
Spec1.02y Partial fractions: decompose rational functions4.06b Method of differences: telescoping series
2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).
Question 2:
AnswerMarks Guidance
Working/AnswerMarks Guidance
\(\frac{2}{r} - \frac{4}{r+1} + \frac{2}{r+2}\)M1A1 Award B2 if written down by cover up rule
\(\left(2-2+\frac{2}{3}\right) + \left(1-\frac{4}{3}+\frac{1}{2}\right) + \ldots + \left(\frac{2}{n-1}-\frac{4}{n}+\frac{2}{n+1}\right) + \left(\frac{2}{n}-\frac{4}{n+1}+\frac{2}{n+2}\right)\)M1A1
\(= 1 - \frac{2}{n+1} + \frac{2}{n+2}\)A1 [5] AEF
Sum to infinity \(= 1\)B1√ [1] 
## Question 2:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\frac{2}{r} - \frac{4}{r+1} + \frac{2}{r+2}$ | M1A1 | Award **B2** if written down by cover up rule |
| $\left(2-2+\frac{2}{3}\right) + \left(1-\frac{4}{3}+\frac{1}{2}\right) + \ldots + \left(\frac{2}{n-1}-\frac{4}{n}+\frac{2}{n+1}\right) + \left(\frac{2}{n}-\frac{4}{n+1}+\frac{2}{n+2}\right)$ | M1A1 | |
| $= 1 - \frac{2}{n+1} + \frac{2}{n+2}$ | A1 [5] | AEF |
| Sum to infinity $= 1$ | B1√ [1] | |

---
2 Express $\frac { 4 } { r ( r + 1 ) ( r + 2 ) }$ in partial fractions and hence find $\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }$.

Deduce the value of $\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }$.

\hfill \mbox{\textit{CAIE FP1 2016 Q2 [6]}}
This paper (13 questions)
View full paper
Q1 4 Q2 6 Q3 6 Q4 8 Q5 9 Q6 9 Q7 10 Q8 11 Q9 11 Q10 12 Q11 EITHER Q11 OR Q15