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8 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 2 · Question 1(c)

Given that S(n) = 5 + 5² + 5³ + 5⁴ + ... + 5ⁿ.

Use mathematical induction to prove that 4S(n) = 5ⁿ⁺¹ – 5 for n ∈ N.

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Other parts of this question

  1. 1(a)(i)State the inverse and the contrapositive of the statement p → q.[2 marks]
  2. 1(a)(ii)Copy and complete the table below to show the truth table for p → q and ~q → ~p.[4 marks]
  3. 1(a)(iii)Hence, state whether the compound statements p ↔ q and ~q → ~p are logically equivalent. Justify your response.[2 marks]
  4. 1(b)(i)Find the values of p and q.[4 marks]
  5. 1(b)(ii)Hence, factorize f(x) = x³ + px² – x + q completely.[5 marks]

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