Mathematical induction — A/L Combined Mathematics

How do I practise Mathematical induction?

1 question with a worked solution

Check before you answer: Explicit Base Step: Writing the direct calculation verifying P(1) explicitly (omitting this forfeits base marks).

1. In proving by the Principle of Mathematical Induction that a proposition P(n) is true for all positive integers n ≥ 1, what are the two mandatory deductive steps required?
   1. A) Differentiate P(n) with respect to n and set the derivative to zero
   2. B) Assume P(n) is true for all n and prove P(0) = 0
   3. C) Base step: Verify P(1) is true; Inductive step: Prove that if P(k) is true, then P(k + 1) is necessarily true
   4. D) Compute numerical values for n = 1, 2, 3, 4, 5 and extrapolate without proof
   Check my answer

   Correct answer: C) Base step: Verify P(1) is true; Inductive step: Prove that if P(k) is true, then P(k + 1) is necessarily true

   Mathematical induction consists of: (1) Basis Step: verifying that P(1) holds; and (2) Inductive Step: showing that for any arbitrary integer k ≥ 1, the assumption that P(k) is true (inductive hypothesis) logically guarantees that P(k + 1) is true. By the induction axiom, P(n) is true for all n ∈ ℤ⁺.

   Show worked solution

   Step 1: Base step: Establish truth of P(1). Step 2: Inductive step: Assume P(k) is true, deduce that P(k+1) is true. Answer: Base step: Verify P(1) is true; Inductive step: Prove that if P(k) is true, then P(k + 1) is necessarily true.
   Modelled on: the A/L Combined Mathematics examination pattern

Get this auto-marked

Next in the syllabus

Where does Mathematical induction fit in the syllabus?

Year 1 · Term 3