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CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2 · Question 3(b)(i)

Use mathematical induction to prove that 1^3 + 2^3 + \ldots + n^3 = \frac{1}{4}n^2(n+1)^2, for n \in \mathbb{N}.

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  1. 3(a)(i)Determine the Taylor series expansion about x = 2 of the function f(x) = \ln(5 + x) up to and including the term in x^3.[6 marks]
  2. 3(a)(ii)Hence, obtain an approximation for f(7) - \ln(7).[2 marks]
  3. 3(b)(ii)Hence, or otherwise, show that \sum_{i=1}^{2n+1} i^3 = (2n + 1)^2(n + 1)^2.[3 marks]
  4. 3(b)(iii)Use the results of Parts (b)(i) and (ii) to show that \sum_{i=1}^{n+1} (2i - 1)^3 = (n + 1)^2(2n^2 + 4n + 1).[5 marks]

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