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7 marksSequences

CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2 · Question 3(b)(i)

The sequence of positive terms, {x_n}, is defined by x_{n+1} = x_n^2 + 1/4, x_1 < 1/2.

Show, by mathematical induction, or otherwise, that x_n < 1/2 for all positive integers n.

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Other parts of this question

  1. 3(a)(i)Show that the terms of sum_{r=1}^m ln 3^r are in arithmetic progression.[3 marks]
  2. 3(a)(ii)Find the sum of the first 20 terms of this series.[4 marks]
  3. 3(a)(iii)Hence, show that sum_{r=1}^{2m} ln 3^r = (2m^2 + m) ln 3.[3 marks]
  4. 3(b)(ii)By considering x_{n+1} - x_n, or otherwise, show that x_n < x_{n+1}.[3 marks]

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