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8 marksSeries

CAPE Pure Mathematics Unit 2 · May/June 2026 · Paper 2 · Question 3(b)(ii)

This section covers series.

Prove by mathematical induction that 4 + 4^2 + 4^3 + ... + 4^n = 4/3 (4^n - 1) for all positive integers n.

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Other parts of this question

  1. 3(a)(i)State the values of the first FOUR terms of the sequence, a_1, a_2, a_3, and a_4.[2 marks]
  2. 3(a)(ii)Show that the sequence converges.[2 marks]
  3. 3(b)(i)Express the series 4 + 4^2 + 4^3 + ... + 4^n using summation notation.[2 marks]
  4. 3(c)(i)Determine the values of A and B such that 4/((2r+1)(2r+3)) = A/(2r+1) + B/(2r+3).[3 marks]
  5. 3(c)(ii)Hence, use the method of differences to show that Σ(from r=1 to n) 4/((2r+1)(2r+3)) = 2(1/3 - 1/(2n+3)).[5 marks]
  6. 3(c)(iii)Hence, calculate Σ(from r=1 to ∞) 4/((2r+1)(2r+3)).[3 marks]

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