CAIE FP1 2012 June — Question 1 5 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeInfinite series convergence and sum
DifficultyStandard +0.8 This requires recognizing the general term 1/(r(r+2)), applying partial fractions to get 1/2(1/r - 1/(r+2)), then telescoping the series to find S_n, and finally taking the limit as nā†’āˆž. While methodical, it demands multiple non-trivial steps including partial fractions and telescoping series manipulation, making it moderately challenging but still within standard Further Maths scope.
Spec4.06b Method of differences: telescoping series
1 Find the sum of the first \(n\) terms of the series $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$ and deduce the sum to infinity.
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{1}{r(r+2)} = \frac{1}{2}\left\{\frac{1}{r} - \frac{1}{r+2}\right\}\)M1A1 Finds partial fractions
\(\frac{1}{2}\left\{\left[\frac{1}{n}-\frac{1}{n+2}\right]+\left[\frac{1}{n-1}-\frac{1}{n+1}\right]+\ldots+\left[\frac{1}{2}-\frac{1}{4}\right]+\left[1-\frac{1}{3}\right]\right\}\)M1 Use of method of differences
\(= \frac{1}{2}\left\{\frac{3}{2}-\frac{1}{n+1}-\frac{1}{n+2}\right\}\) (acf) \(\Rightarrow S_\infty = \frac{3}{4}\)A1A1\(\sqrt{}\) Obtains result; Part Mark: 5 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{r(r+2)} = \frac{1}{2}\left\{\frac{1}{r} - \frac{1}{r+2}\right\}$ | M1A1 | Finds partial fractions |
| $\frac{1}{2}\left\{\left[\frac{1}{n}-\frac{1}{n+2}\right]+\left[\frac{1}{n-1}-\frac{1}{n+1}\right]+\ldots+\left[\frac{1}{2}-\frac{1}{4}\right]+\left[1-\frac{1}{3}\right]\right\}$ | M1 | Use of method of differences |
| $= \frac{1}{2}\left\{\frac{3}{2}-\frac{1}{n+1}-\frac{1}{n+2}\right\}$ (acf) $\Rightarrow S_\infty = \frac{3}{4}$ | A1A1$\sqrt{}$ | Obtains result; Part Mark: 5 |

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1 Find the sum of the first $n$ terms of the series

$$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$

and deduce the sum to infinity.

\hfill \mbox{\textit{CAIE FP1 2012 Q1 [5]}}
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Q1 5 Q2 5 Q3 8 Q4 8 Q5 9 Q6 9 Q7 10 Q8 10 Q9 11 Q10 11 Q11 EITHER Q11 OR