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5 marksSequences

CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 3(a)(i)

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

Show, by mathematical induction, that x_n < \frac{1}{2} for all positive integers n.

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Other parts of this question

  1. 3(a)(ii)By considering x_{n+1} - x_n, show that x_n < x_{n+1}.[3 marks]
  2. 3(b)(i)Find the constants A and B such that \frac{2 - 3x}{(1 - x)(1 - 2x)} \equiv \frac{A}{1 - x} + \frac{B}{1 - 2x}.[3 marks]
  3. 3(b)(ii)Obtain the first FOUR non-zero terms of the expansion of each of (1 - x)^{-1} and (1 - 2x)^{-1} as power series of x in ascending order.[4 marks]
  4. 3(b)(iii) a)Find the range of values of x for which the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)} is valid.[2 marks]
  5. 3(b)(iii) b)Find the coefficient of x^n in the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)}.[2 marks]
  6. 3(iv)The sum, S_n, of the first n terms of a series is given by S_n = n(3n - 4). Show that the series is an Arithmetic Progression (A.P.) with common difference 6.[6 marks]

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