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4 marksSeries

CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 4(a)(i)

A Geometric Progression (G.P.) with first term a and common ratio r, 0 < r < 1, is such that the sum of the first three terms is \frac{26}{3} and their product is 8.

Show that r + 1 + \frac{1}{r} = \frac{13}{3}.

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Other parts of this question

  1. 4(a)(ii) a)Hence, find the value of r.[4 marks]
  2. 4(a)(ii) b)Find the value of a.[1 mark]
  3. 4(a)(ii) c)Find the sum to infinity of the G.P.[2 marks]
  4. 4(b)Expand \frac{2}{e^x + e^{-x}}, |x| < 1 in ascending powers of x as far as the term in x^4.[5 marks]
  5. 4(c)(i)Express f(r) - f(r + 1) in terms of r.[3 marks]
  6. 4(c)(ii)Hence, or otherwise, find S_n = \sum_{r=1}^n \frac{3}{r(r + 1)(r + 2)}.[4 marks]
  7. 4(c)(iii)Deduce the sum to infinity of the series in (c)(ii).[2 marks]

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