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Sequences · CAPE Pure Mathematics Unit 2

45 past-paper questions on Sequences, part of Sequences, Series and Approximations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Sequences
  1. 6(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Obtain an expression for the number of visitors on the nth day.
  2. 1(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2What is the LARGEST number reached by the membership of the club?
  3. 1(b)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Calculate the EXACT value of k and of r.
  4. 1(b)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2How many members will there be in the club 3 years after its formation?
  5. 3(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Show, by mathematical induction, or otherwise, that x_n < 1/2 for all positive integers n.
  6. 3(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2By considering x_{n+1} - x_n, or otherwise, show that x_n < x_{n+1}.
  7. 3(a)9 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Prove by Mathematical Induction that u_n = n! for all n \in \mathbb{N}.
  8. 3(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2A sequence \{u_n\} is defined by the recurrence relation u_{n+1} = u_n + n, u_1 = 3, n \in \mathbb{N}. State the first FOUR terms of the sequence.
  9. 3(a)(ii)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Prove by mathematical induction, or otherwise, that u_n = \frac{n^2 - n + 6}{2}.
  10. 4(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Find a_2 and a_3.
  11. 4(a)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Express a_{n+1} - 2 in terms of a_n.
  12. 4(a)(iii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that a_{n+1} < 2.
  13. 4(a)(iii)b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that a_n < a_{n+1}.
  14. 3(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Determine t_2, t_3 and t_4.
  15. 3(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Express t_n in terms of n.
  16. 1(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Determine the initial temperature of the water in the tank.
  17. 1(a)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Determine the temperature at which the water in the tank will eventually stabilize.
  18. 3(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Express, in terms of r, the r^{\text{th}} term of the sequence.
  19. 3(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2The 9^{\text{th}} term of an A.P. is three times the 3^{\text{rd}} term and the sum of the first 10 terms is 110. Find the first term a and the common difference d.
  20. 3(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show, by mathematical induction, that x_n < \frac{1}{2} for all positive integers n.
  21. 3(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2By considering x_{n+1} - x_n, show that x_n < x_{n+1}.
  22. 3(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Calculate the first term, a, and the common ratio, r.
  23. 3(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Express, in terms of r, the r^{\text{th}} term, u_r, of the sequence.
  24. 3(a)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2The sequence \{a_n\} is defined by a_1 = 1, a_{n+1} = 4 + 2\sqrt[3]{a_n}. Use mathematical induction to prove that 1 \le a_n \le 8 for all n in the set of positive integers.
  25. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If the terms of the sequence u_1, u_2, u_3, \dots, u_n, \dots satisfy the recurrence relation u_{n+1} = u_n + 3, n \ge 1, then the n^{\text{th}} term may be expressed as
  26. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The 5^{\text{th}} term in the sequence that is defined by the relation u_n = (-1)^{n+1}\frac{n}{3n - 1}, n \ge 1, is
  27. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Which of the following sequences, \{u_n\}, converges?
  28. 3(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Determine \lim_{n \to \infty} T_n.
  29. 3(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Show that T_4 = \frac{9}{4} \left(1 + \frac{1}{16}\right)^{-\frac{1}{2}}.
  30. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If the terms of the sequence u_1, u_2, u_3, \dots, u_n, \dots satisfy the recurrence relation u_{n+1} = u_n + 3, n \ge 1, then the n^{\text{th}} term may be expressed as
  31. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1Given that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
  32. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1For the recurrence relation a_{n+1} = a_n - a_{n-1} where a_1 = 1 and a_2 = 3, the value of a_5 is
  33. 3(a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Given that u_8 = 13x + 1 and that u_10 = 34x + 1, find (u_9)'.
  34. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
  35. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1A sequence is defined as u_{n+1} = 1 - \frac{1}{1 + u_n}, where u_1 = 1 and n \in \mathbb{N}. The 20th term of the sequence is
  36. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Given that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
  37. 3(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2State the third term, a_3, of the sequence.
  38. 3(a)(ii)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Use mathematical induction to prove that a_n is increasing and bounded above by 3, so a_n < a_(n+1) and a_n <= 3 for all n in N.
  39. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
  40. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1A sequence is defined as u_{n+1} = 1 - \frac{1}{1 + u_n} where u_1 = 1 and n \in \mathbb{N}. The 20th term of the sequence is
  41. 3(a)(i)2 marks· Pure Mathematics · Unit 2 Q3 3(a)(i)State the values of the first FOUR terms of the sequence, a_1, a_2, a_3, and a_4.
  42. 3(a)(ii)2 marks· Pure Mathematics · Unit 2 Q3 3(a)(ii)Show that the sequence converges.
  43. 3(c)4 marks· Pure Mathematics · Unit 2 Q3 3(c)The first and fifth terms of a geometric progression are 16 and 9, respectively. Determine the THIRD term of the progression.
  44. 4(b)(i)4 marks· Pure Mathematics · Unit 2 Q4 4(b)(i)Write down an expression for the nᵗʰ term of the sequence.
  45. 4(b)(ii)3 marks· Pure Mathematics · Unit 2 Q4 4(b)(ii)Determine whether the sequence converges or diverges.

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