Edexcel CP1 Specimen — Question 1 5 marks

Exam BoardEdexcel
ModuleCP1 (Core Pure 1)
SessionSpecimen
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeMethod of differences with given identity
DifficultyStandard +0.8 This is a Core Pure 1 question requiring method of differences with partial fractions, algebraic manipulation to sum the telescoping series, and then matching coefficients to find constants a and b. While the technique is standard for Further Maths, the multi-step algebraic manipulation and need to simplify to the given form makes it moderately challenging, placing it above average difficulty.
Spec4.06b Method of differences: telescoping series
  1. Prove that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$ where \(a\) and \(b\) are constants to be found.
Question 1:
Main Scheme:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Valid attempt at partial fractionsM1
Starts the process of differences to identify relevant fractions at start and endM1
Correct fractions that do not cancelA1
Attempt common denominatorM1
Correct answerA1
Alternative by Induction:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Uses \(n=1\) and \(n=2\) to identify values for \(a\) and \(b\)M1
Starts induction by adding the \((k+1)\)th term to sum of \(k\) termsM1
Correct single fractionA1
Attempt to factorise the numeratorM1
Correct answer and conclusionA1 
## Question 1:

**Main Scheme:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Valid attempt at partial fractions | M1 | |
| Starts the process of differences to identify relevant fractions at start and end | M1 | |
| Correct fractions that do not cancel | A1 | |
| Attempt common denominator | M1 | |
| Correct answer | A1 | |

**Alternative by Induction:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $n=1$ and $n=2$ to identify values for $a$ and $b$ | M1 | |
| Starts induction by adding the $(k+1)$th term to sum of $k$ terms | M1 | |
| Correct single fraction | A1 | |
| Attempt to factorise the numerator | M1 | |
| Correct answer and conclusion | A1 | |

---
\begin{enumerate}
  \item Prove that
\end{enumerate}

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$

where $a$ and $b$ are constants to be found.\\

\hfill \mbox{\textit{Edexcel CP1  Q1 [5]}}
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