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CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2 · Question 3(b)(ii)

The first four terms of a sequence are 1 \times 3, 2 \times 4, 3 \times 5, 4 \times 6.

Prove, by mathematical induction, that \sum_{r=1}^n u_r = \frac{1}{6}n(n + 1)(2n + 7), \forall n \in \mathbb{N}.

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Other parts of this question

  1. 3(a)(i)Calculate the first term, a, and the common ratio, r.[5 marks]
  2. 3(a)(ii)Hence, calculate n if S_n = 177\,146.[4 marks]
  3. 3(b)(i)Express, in terms of r, the r^{\text{th}} term, u_r, of the sequence.[2 marks]
  4. 3(c)(i)Use Maclaurin's Theorem to find the first three non-zero terms in the power series expansion of \cos 2x.[5 marks]
  5. 3(c)(ii)Hence, or otherwise, obtain the first two non-zero terms in the power series expansion of \sin^2 x.[2 marks]

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