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CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2 · Question 3(a)(ii)

Let S = \sum_{r=1}^n r and T = \sum_{r=1}^n (n + 1 - r).

Use the principle of mathematical induction to prove that \sum_{r=1}^n r^2 = \frac{1}{6} n(n + 1)(2n + 1).

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Other parts of this question

  1. 3(a)(i)a)Show that T = S.[2 marks]
  2. 3(a)(i)b)Deduce that S = \frac{1}{2} n(n + 1).[3 marks]
  3. 3(a)(iii)Hence, prove that \sum_{r=1}^n 2r(3r + 1) = 2n(n + 1)^2.[4 marks]
  4. 3(b)(i)Show that the equation x^3 + 3x^2 + 6x - 3 = 0 has a root \alpha between 0 and 1.[2 marks]
  5. 3(b)(ii)Prove that \alpha is the only real root.[3 marks]
  6. 3(b)(iii)Using TWO iterations of the Newton-Raphson method, find \alpha correct to 2 decimal places.[4 marks]

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