Common mistake: Writing "assume the result is true for n = k + 1"

Writing "assume the result is true for n = k + 1" — you assume for n = k, then prove for n = k + 1

Common mistake: Weak inductive step that adds the (k+1)th term without using the inductive assumption

Weak inductive step that adds the (k+1)th term without using the inductive assumption

Common mistake: In contradiction proofs, not clearly stating the assumption at the start

In contradiction proofs, not clearly stating the assumption at the start

Common mistake: Missing the conclusion sentence

Missing the conclusion sentence — always end with "therefore, by the principle of mathematical induction..."

VCE Units 1–4 · Specialist Maths

VCE Specialist Maths — Proof

Mathematical proof is a core skill in Specialist Mathematics. Unlike other topics where an answer is sufficient, proof requires a logically complete argument that convinces any reader. The two main methods assessed in Specialist are proof by mathematical induction and proof by contradiction. Both have a fixed structure that must be followed exactly.

Key Concepts & Formulas

Proof by induction — four steps: (1) Base case: verify the statement holds for n = 1, (2) Inductive assumption: assume it holds for n = k, (3) Inductive step: use the assumption to prove it holds for n = k + 1, (4) Conclusion: by induction, the statement holds for all n ≥ 1
The inductive step must use the assumption explicitly — you cannot assume k + 1 without using k
Common induction targets: sum formulas (e.g. 1 + 2 + ... + n = n(n+1)/2), divisibility (e.g. 3ⁿ − 1 is divisible by 2), inequality proofs
Proof by contradiction: assume the negation of what you want to prove, then derive a logical contradiction
Common contradiction targets: irrationality proofs (√2 is irrational), infinitely many primes
Direct proof: start from given conditions and apply known results step by step to reach the conclusion
Proof by contrapositive: "if P then Q" is equivalent to "if not Q then not P"
All steps in a proof must follow logically — never assume the result you're trying to prove

Practice Questions

4 questions

Attempt each question before reading the hint. These are styled to match VCE exam format.

Q1.Prove by mathematical induction that 1 + 2 + 3 + ... + n = n(n+1)/2 for all positive integers n.

4 marks

Q2.Use mathematical induction to prove that 4ⁿ − 1 is divisible by 3 for all n ≥ 1.

4 marks

Q3.Prove that √3 is irrational.

3 marks

Show hint

Assume √3 = p/q in lowest terms and derive a contradiction.

Q4.Prove by induction that for all n ≥ 1: 1·2 + 2·3 + ... + n(n+1) = n(n+1)(n+2)/3.

5 marks

Common Mistakes to Avoid

These are the errors that VCE students most frequently make in Proof — and that examiners are specifically watching for.

Writing "assume the result is true for n = k + 1" — you assume for n = k, then prove for n = k + 1
Weak inductive step that adds the (k+1)th term without using the inductive assumption
In contradiction proofs, not clearly stating the assumption at the start
Missing the conclusion sentence — always end with "therefore, by the principle of mathematical induction..."

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