Question 5 - A-Level Maths

Edexcel FP2 2013 June — Question 5 7 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2013
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Partial Fractions
Type	Three linear factors in denominator
Difficulty	Standard +0.8 This is a standard Further Maths partial fractions question with three linear factors, followed by a telescoping series proof. Part (a) is routine FP2 content, but part (b) requires careful bookkeeping with the method of differences and algebraic manipulation to reach the given form. The multi-step nature and need to recognize the telescoping pattern elevates this above average difficulty.
Spec	4.06b Method of differences: telescoping series

5. (a) Express $\frac { 2 } { r ( r + 1 ) ( r + 2 ) }$ in partial fractions.
(b) Using your answer to part (a) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( n + 3 ) } { 2 ( n + 1 ) ( n + 2 ) }$$

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Question 5:

Part (a):

Answer	Marks	Guidance
$\frac{1}{r} - \frac{2}{(r+1)} + \frac{1}{(r+2)}$	M1 A1 A1	(3 marks)

Part (b):

Answer	Marks	Guidance
Write out LHS as telescoping sum beginning $\frac{1}{1} - \frac{2}{2} + \frac{1}{3} + \frac{1}{2} - \frac{2}{3} + \frac{1}{4} + \ldots$	M1
Complete final three rows shown: $+\frac{1}{(n-2)} - \frac{2}{(n-1)} + \frac{1}{n}$, $+\frac{1}{(n-1)} - \frac{2}{n} + \frac{1}{(n+1)}$, $+\frac{1}{n} - \frac{2}{(n+1)} + \frac{1}{(n+2)}$	A1
$= 1 - \frac{1}{2} - \frac{1}{n+1} + \frac{1}{n+2} = \frac{n^2+3n+2-2n-4+2n+2}{2(n+1)(n+2)}$	M1 dep
$= \frac{n^2+3n}{2(n+1)(n+2)} = \frac{n(n+3)}{2(n+1)(n+2)}$ *	A1	(4 marks)

## Question 5:

### Part (a):
| $\frac{1}{r} - \frac{2}{(r+1)} + \frac{1}{(r+2)}$ | M1 A1 A1 | (3 marks) |

### Part (b):
| Write out LHS as telescoping sum beginning $\frac{1}{1} - \frac{2}{2} + \frac{1}{3} + \frac{1}{2} - \frac{2}{3} + \frac{1}{4} + \ldots$ | M1 | |
| Complete final three rows shown: $+\frac{1}{(n-2)} - \frac{2}{(n-1)} + \frac{1}{n}$, $+\frac{1}{(n-1)} - \frac{2}{n} + \frac{1}{(n+1)}$, $+\frac{1}{n} - \frac{2}{(n+1)} + \frac{1}{(n+2)}$ | A1 | |
| $= 1 - \frac{1}{2} - \frac{1}{n+1} + \frac{1}{n+2} = \frac{n^2+3n+2-2n-4+2n+2}{2(n+1)(n+2)}$ | M1 dep | |
| $= \frac{n^2+3n}{2(n+1)(n+2)} = \frac{n(n+3)}{2(n+1)(n+2)}$ * | A1 | (4 marks) |

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5. (a) Express $\frac { 2 } { r ( r + 1 ) ( r + 2 ) }$ in partial fractions.\\
(b) Using your answer to part (a) and the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( n + 3 ) } { 2 ( n + 1 ) ( n + 2 ) }$$

\hfill \mbox{\textit{Edexcel FP2 2013 Q5 [7]}}

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Q1 7 Q2 9 Q3 9 Q4 9 Q5 7 Q6 8 Q7 7 Q8 8 Q9 11

Answer	Marks	Guidance
Write out LHS as telescoping sum beginning \(\frac{1}{1} - \frac{2}{2} + \frac{1}{3} + \frac{1}{2} - \frac{2}{3} + \frac{1}{4} + \ldots\)	M1
Complete final three rows shown: \(+\frac{1}{(n-2)} - \frac{2}{(n-1)} + \frac{1}{n}\), \(+\frac{1}{(n-1)} - \frac{2}{n} + \frac{1}{(n+1)}\), \(+\frac{1}{n} - \frac{2}{(n+1)} + \frac{1}{(n+2)}\)	A1
\(= 1 - \frac{1}{2} - \frac{1}{n+1} + \frac{1}{n+2} = \frac{n^2+3n+2-2n-4+2n+2}{2(n+1)(n+2)}\)	M1 dep
\(= \frac{n^2+3n}{2(n+1)(n+2)} = \frac{n(n+3)}{2(n+1)(n+2)}\) *	A1	(4 marks)