OCR Further Pure Core 1 2021 November — Question 10 8 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2021
SessionNovember
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeFinding n for given sum value
DifficultyChallenging +1.2 This is a Further Maths question requiring partial fractions decomposition, telescoping series recognition, and solving an inequality. While it involves multiple steps (decompose, sum telescoping series, solve for n), the techniques are standard for FP1 level and the telescoping pattern is a common textbook exercise. The 8 marks reflect length rather than exceptional difficulty.
Spec4.06b Method of differences: telescoping series
Using an algebraic method, determine the least value of \(n\) for which \(\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49\). [8]
Question 10:
AnswerMarks
101 A B
= +
(2r−1)(2r+1) 2r−1 2r+1
⇒ A(2r+1)+B(2r−1)=1
⇒ A−B=1, A+B=0
1 1
⇒ A= ,B=−
2 2
1 1 1 1 
 − + − +... 
n 1 1 1 3 3 5
⇒∑ =  
(2r−1)(2r+1) 2  1 1   1 1 
r=1 + − + − 
 2n−3 2n−1 2n−1 2n+1
1 1 
= 1−  oe
2 2n+1
1 1 
1− ≥0.49
2 2n+1
⇒n≥0.98n+0.49
0.49
⇒n≥ =24.5
0.02
AnswerMarks
⇒n=25M1
M1
A1
M1
M1
A1
M1
AnswerMarks
A13.1a
1.1
1.1
3.1a
2.1
1.1
3.1a
AnswerMarks
3.2apartial fractions
Allow any method to determine A and B
Both values
Use of differences
Deal with subtraction
Use of inequality on their formula
No marks for a purely numerical solution.
[8]
Question 10:
10 | 1 A B
= +
(2r−1)(2r+1) 2r−1 2r+1
⇒ A(2r+1)+B(2r−1)=1
⇒ A−B=1, A+B=0
1 1
⇒ A= ,B=−
2 2
1 1 1 1 
 − + − +... 
n 1 1 1 3 3 5
⇒∑ =  
(2r−1)(2r+1) 2  1 1   1 1 
r=1 + − + − 
 2n−3 2n−1 2n−1 2n+1
1 1 
= 1−  oe
2 2n+1
1 1 
1− ≥0.49
2 2n+1
⇒n≥0.98n+0.49
0.49
⇒n≥ =24.5
0.02
⇒n=25 | M1
M1
A1
M1
M1
A1
M1
A1 | 3.1a
1.1
1.1
3.1a
2.1
1.1
3.1a
3.2a | partial fractions
Allow any method to determine A and B
Both values
Use of differences
Deal with subtraction
Use of inequality on their formula
No marks for a purely numerical solution.
[8]
Using an algebraic method, determine the least value of $n$ for which $\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49$. [8]

\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q10 [8]}}
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Q1 6 Q2 8 Q3 8 Q4 11 Q5 4 Q6 3 Q7 9 Q8 8 Q9 5 Q10 8 Q11 5