OCR Further Additional Pure 2023 June — Question 4 7 marks

Exam BoardOCR
ModuleFurther Additional Pure (Further Additional Pure)
Year2023
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and Series
TypeConvergence and Limits of Sequences
DifficultyHard +2.3 This is a challenging Further Maths question requiring reduction formula techniques for the integral, manipulation of recurrence relations to prove monotonicity, and limit analysis. It demands sophisticated proof skills and insight into how the ratio behaves as n→∞, going well beyond standard A-level material.
Spec8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic
4 The sequence \(\left\{ A _ { n } \right\}\) is given for all integers \(n \geqslant 0\) by \(A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }\), where \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x\).
  • Show that \(\left\{ A _ { n } \right\}\) increases monotonically.
  • Show that \(\left\{ \mathrm { A } _ { \mathrm { n } } \right\}\) converges to a limit, \(A\), whose exact value should be stated.
Question 4:
AnswerMarks
4𝐼 = ∫𝑐𝑜𝑠𝑛+1𝑥 . 𝑐𝑜𝑠𝑥 d𝑥 = [𝑐𝑜𝑠𝑛+1𝑥 𝑠𝑖𝑛𝑥] ...
𝑛+2
−∫(𝑛+1)𝑐𝑜𝑠𝑛𝑥(−𝑠𝑖𝑛𝑥) . 𝑠𝑖𝑛𝑥 d𝑥
= (0 +) (n +1)(I n – I n+2)
 (n+2) I n = (n + 1) I n +2
I n+1 1
Then A = n + 2 = or 1−
n I n+2 n+2
n
→ 1 as n →  i.e. A = 1
1 1
Since A = 1 −  1 − = A (for all n)
n + 1 n + 3 n + 2 n
 
it follows that A is monotonic increasing
AnswerMarks
nM1*
A1
M1dep
A1
B1
M1
A1
AnswerMarks
[7]3.1a
1.1
2.1
1.1
2.2a
2.1
AnswerMarks
3.2aCorrect use of integration by parts. Allow sign errors
in der/int of cos𝑥.
Or starting from 𝐼
𝑛
Fully correct first stage
Use of s2 = 1 – c2 to express the integral(s) correctly
in terms of I k and limits substituted.
Correct reduction formula
SC1 for arriving at correct 𝐴 with correct evidence
𝑛
for at least three consecutive terms e.g.
1 2 3
𝐴 = ,𝐴 = ,𝐴 = .
0 1 2
2 3 4
From consideration of the relevant ratio of terms
1
Or considering A − A =  0
n + 1 n (n+2)(n+3)
𝐴𝑛+1
Or considering > 1 oe
𝐴𝑛
Condone ≥
Correct answer from appropriate reasoning
AnswerMarks Guidance
(a)i  1 9 17 25
3 2
1 1 9 17 25
9 9 17 25 1
17 17 25 1 9
AnswerMarks
25 25 1 9 17B1
B1
AnswerMarks
[2]1.1
1.1Any two bold entries (in shaded squares) correct
All bold entries (in shaded squares) correct
AnswerMarks
iiG  C
4
Since either G has a generator (9 or 25)
or from the main diagonal, only 2 of G’s elements
are self-inverse, so G ≇ K
AnswerMarks
4B1
B1
AnswerMarks
[2]2.2a
2.4Allow statement that G is cyclic
Either generator noted
Or statement that only 1 non-identity element is self-
inverse
AnswerMarks Guidance
i1, 9, 25, 17 B1
B1
AnswerMarks
[2]1.2
1.1Any one noted (other than 1)
All four noted (and no extras)
Question 4:
4 | 𝐼 = ∫𝑐𝑜𝑠𝑛+1𝑥 . 𝑐𝑜𝑠𝑥 d𝑥 = [𝑐𝑜𝑠𝑛+1𝑥 𝑠𝑖𝑛𝑥] ...
𝑛+2
−∫(𝑛+1)𝑐𝑜𝑠𝑛𝑥(−𝑠𝑖𝑛𝑥) . 𝑠𝑖𝑛𝑥 d𝑥
= (0 +) (n +1)(I n – I n+2)
 (n+2) I n = (n + 1) I n +2
I n+1 1
Then A = n + 2 = or 1−
n I n+2 n+2
n
→ 1 as n →  i.e. A = 1
1 1
Since A = 1 −  1 − = A (for all n)
n + 1 n + 3 n + 2 n
 
it follows that A is monotonic increasing
n | M1*
A1
M1dep
A1
B1
M1
A1
[7] | 3.1a
1.1
2.1
1.1
2.2a
2.1
3.2a | Correct use of integration by parts. Allow sign errors
in der/int of cos𝑥.
Or starting from 𝐼
𝑛
Fully correct first stage
Use of s2 = 1 – c2 to express the integral(s) correctly
in terms of I k and limits substituted.
Correct reduction formula
SC1 for arriving at correct 𝐴 with correct evidence
𝑛
for at least three consecutive terms e.g.
1 2 3
𝐴 = ,𝐴 = ,𝐴 = .
0 1 2
2 3 4
From consideration of the relevant ratio of terms
1
Or considering A − A =  0
n + 1 n (n+2)(n+3)
𝐴𝑛+1
Or considering > 1 oe
𝐴𝑛
Condone ≥
Correct answer from appropriate reasoning
(a) | i |  1 9 17 25
3 2
1 1 9 17 25
9 9 17 25 1
17 17 25 1 9
25 25 1 9 17 | B1
B1
[2] | 1.1
1.1 | Any two bold entries (in shaded squares) correct
All bold entries (in shaded squares) correct
ii | G  C
4
Since either G has a generator (9 or 25)
or from the main diagonal, only 2 of G’s elements
are self-inverse, so G ≇ K
4 | B1
B1
[2] | 2.2a
2.4 | Allow statement that G is cyclic
Either generator noted
Or statement that only 1 non-identity element is self-
inverse
i | 1, 9, 25, 17 | B1
B1
[2] | 1.2
1.1 | Any one noted (other than 1)
All four noted (and no extras)
4 The sequence $\left\{ A _ { n } \right\}$ is given for all integers $n \geqslant 0$ by $A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }$, where $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x$.

\begin{itemize}
  \item Show that $\left\{ A _ { n } \right\}$ increases monotonically.
  \item Show that $\left\{ \mathrm { A } _ { \mathrm { n } } \right\}$ converges to a limit, $A$, whose exact value should be stated.
\end{itemize}

\hfill \mbox{\textit{OCR Further Additional Pure 2023 Q4 [7]}}
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