Mathematical induction — A/L Combined Mathematics

The Principle of Mathematical Induction is a rigorous deductive proof method for statements P(n) asserted for all positive integers n ∈ ℤ⁺. The proof consists of three mandatory logical stages: (1) Base Step: Proving P(1) is true directly by evaluation; (2) Inductive Hypothesis: Assuming P(k) is true for an arbitrary positive integer k; (3) Inductive Step: Demonstrating that the truth of P(k) strictly implies the truth of P(k + 1). Once both steps are proved, by mathematical induction P(n) is true for all n ∈ ℤ⁺. This technique is applied to summation identities, matrix powers, inequalities, and divisibility proofs.

Check before you answer (A/L Pure Maths Paper 1)

1. Explicit Base Step: Writing the direct calculation verifying P(1) explicitly (omitting this forfeits base marks).
2. Clear Hypothesis Statement: Explicitly stating 'Assume P(k) is true for some k ∈ ℤ⁺' rather than vague wording.
3. Inductive Step Algebraic Rigour: Showing the algebraic addition of the (k + 1)-th term to the assumed sum S_k and factorizing clearly to reveal the form of S_k+1.