Intro
Mathematical induction proves a statement P(n) for all integers n ≥ n0 by verifying a base case and showing that P(k) implies P(k + 1). In HSC Extension 2, induction appears on divisibility, inequalities, and sums — marks are awarded for clear structure, not clever algebra alone. For NSW Year 12 students following the NESA syllabus. Keywords: induction proof method, HSC Extension 2 induction, step-by-step math proof.
Summary
Induction has four labelled parts: base case, inductive assumption, inductive step, and conclusion. Divisibility proofs rewrite f(k + 1) in terms of f(k); inequality proofs often compare successive terms or invoke a known bound. Write each step as a sentence examiners can tick.
Induction sits in the Extension 2 proof strand alongside direct proof and contradiction. Past papers have tested divisibility, inequalities, and sums of powers — recognise which type you face before writing the base case.
Key Points
- Base case: verify P(n0) directly — usually n = 1 or n = 0.
- Inductive assumption: assume P(k) for an arbitrary k ≥ n0; do not assume P(k + 1).
- Inductive step: prove P(k + 1) using the assumption — this is where marks are won or lost.
- Conclusion: state 'By mathematical induction, P(n) holds for all n ≥ n0.'
- Divisibility: express f(k + 1) = f(k) + (multiple of d) or factor using the assumption.
- Use the HSC Induction booklet for typed examples at exam difficulty.
Worked example
Question. Prove by induction that 3n − 1 is divisible by 2 for all integers n ≥ 1.
Solution.
- Base case (n = 1): 31 − 1 = 2, which is divisible by 2. ✓
- Inductive assumption: Assume 3k − 1 is divisible by 2 for some k ≥ 1; write 3k − 1 = 2m for integer m.
- Inductive step: Consider 3k+1 − 1 = 3(3k) − 1 = 3(3k − 1) + 3 − 1 = 3(2m) + 2 = 2(3m + 1).
- Since 3m + 1 is an integer, 3k+1 − 1 is divisible by 2.
- Conclusion: By mathematical induction, 3n − 1 is divisible by 2 for all n ≥ 1.
Answer. Proved.
Takeaway. Factor to expose the assumed expression — here 3k − 1 — then show the remainder is divisible by the target.
Exam Preparation
Induction questions in Extension 2 often sit in the middle-to-late band of difficulty. Practise writing proofs from memory with correct headings, then check against model solutions. Pair induction revision with the Proofs booklet for direct and contradiction proofs that share the same logical discipline.
- Learn the template. Write the four-step skeleton until it is automatic.
- Classify the problem. Before starting, decide: divisibility, inequality, or sum — each has a standard opening move.
- Timed proof attempts. Do two induction proofs per week under 12 minutes, marking yourself on structure first.
Strong induction answers read like a template filled with algebra. Examiners penalise proofs that leap from the inductive assumption to the conclusion without showing how k + 1 was handled. For sum proofs, write S(k + 1) − S(k) explicitly. For inequality proofs, compare consecutive terms or cite a known bound such as (1 + x)n ≥ 1 + nx for x ≥ −1. Keep a checklist on your desk: base, assume, step, conclude — tick each before moving on.
Mini-FAQ
Can I skip the inductive assumption label?
No — exam markers expect the phrase 'Assume true for n = k' or equivalent. Skipping it costs communication marks even if the algebra is correct.
What if the base case fails at n = 1?
Choose the smallest n where P(n) is true and start there. Some statements begin at n = 0 or n = 2; read the question carefully.
How is induction different from a direct proof?
Induction proves infinitely many cases via one chain of logic. Direct proofs handle a single universal statement without the k → k + 1 step.
Common mistakes to avoid
- Assuming P(k + 1) in the inductive step instead of deriving it from P(k).
- Omitting the conclusion sentence that names mathematical induction.
- Base case verified for the wrong starting value of n.
- In divisibility proofs, failing to show the remainder is an integer multiple of d.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Induction booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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