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8 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2 · Question 1(c)

Prove, by mathematical induction, that for all natural numbers n, 1 + 5 + 9 + 13 + ... + (4n - 3) = n(2n - 1).

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Other parts of this question

  1. 1(a)(i)Complete the truth table for p, q, r, p ⇒ q, q ⇒ r, (p ⇒ q) ∧ (q ⇒ r), p ∨ q, and (p ∨ q) ⇒ r.[4 marks]
  2. 1(a)(ii)Determine whether (p ⇒ q) ∧ (q ⇒ r) is logically equivalent to (p ∨ q) ⇒ r. Justify your response.[2 marks]
  3. 1(b)Rationalize the denominator of (√3 + √7) / (√5 + √2), expressing your answer in surd form.[3 marks]
  4. 1(d)(i)Given that x + 3 is a factor of x³ - 7x + q, determine the value of q.[2 marks]
  5. 1(d)(ii)Hence, factorize the expression x³ - 7x + q.[4 marks]
  6. 1(e)Determine the remainder when x⁴ - 2x² + x + 1 is divided by 2x - 1.[2 marks]

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