Proof by induction is part of the AA HL course, alongside proof by contradiction and the use of counterexamples. Many students fear it because it looks like "writing", not maths. In fact, induction is one of the most reliable sources of marks in HL: the structure is always the same, and examiners award marks for each step.
The four-step structure
Every induction proof has four parts. Write all four, every time.
- Basis step. Show the statement is true for the first value, usually n = 1. Show both sides separately – don't just write "true".
- Assumption. "Assume the statement is true for n = k", and write out what that means.
- Inductive step. Using the assumption, show the statement is true for n = k + 1. Write the target expression first, so you know where you are heading.
- Conclusion. A full sentence: "Since the statement is true for n = 1, and if true for n = k it is true for n = k + 1, it is true for all n ∈ ℤ⁺ by mathematical induction."
The conclusion carries a mark of its own – and it is the mark students most often throw away.
The three common types
1. Series (sums)
Prove that a sum equals a formula. In the inductive step, write the sum to k + 1 as "the sum to k, plus the (k + 1)th term", then use the assumption and simplify to the formula with k + 1.
2. Divisibility
Prove an expression is divisible by a number, for example 5ⁿ − 1 by 4. Write the assumption as "5ᵏ − 1 = 4m for some integer m", then rewrite the k + 1 expression in terms of 5ᵏ and substitute.
3. Derivatives and other patterns
Prove a formula for the nth derivative, or results like De Moivre's theorem. Differentiate (or multiply) the assumed result once, then show it matches the formula for k + 1.
What examiners look for
- "Assume true for n = k" – not "let n = k" or "assume true for all n" (which assumes what you are proving).
- Clear use of the assumption – mark where you substitute it.
- Algebra that reaches the target form, step by step. If the answer is "given", every step must be shown.
- The full concluding sentence, including the basis and the inductive step.
Common mistakes
- Writing the basis step as "1 = 1 true" without showing both sides.
- Assuming what needs to be proved.
- Losing track of the target – writing the k + 1 form at the start prevents this.
- Algebra errors with indices and factorials.
- Leaving out the conclusion, or writing a vague one.
How to practise
- Learn the template until it is automatic.
- Do one of each type every week in Year 2 – they take ten minutes each.
- Check against the markscheme to see exactly where each mark sits.
- Strengthen algebra – most failed induction proofs fail in the algebra, not the logic.
Note for future students: in the new course (first exams May 2029), proof by counterexample leaves AA HL; proof by induction stays.
How Maths Bodhi helps
Proof is one of the clearest examples of IB marking rewarding communication, not just correct maths. We teach it as a skill:
- The four-step template practised on all three question types.
- Algebra repair for indices, factorials and factorising – where proofs usually break.
- Past IB proof questions marked line by line against the markscheme.
- Links to complex numbers (De Moivre) and calculus, where induction appears in longer questions.
Ajay Vatsyayan (M.Tech, 14+ years teaching maths) teaches IB DP maths at Maths Bodhi and has taught students from schools across Gurgaon, including The Shri Ram School (Moulsari), GD Goenka World School, Suncity School and Shiv Nadar School. Maths Bodhi is independent and not affiliated with any school. Read Ajay's profile, or see our AA HL tutor and IB algebra tutor pages.
Home sessions across Gurgaon – including Golf Course Road and Sector 56 – or online. Book a free demo class.
Related: HL complex numbers, HL Paper 3 and the IB DP maths guide for Gurgaon families.
Course content checked against the current IB AA guide and the 2029 update notes by the Maths Bodhi team (October 2026). Method reviewed by Ajay Vatsyayan.
Frequently asked questions
- Is proof by induction in IB maths SL?
- No. Proof by induction is part of AA HL. SL students meet simple deductive proof but not induction.
- What are the steps of proof by induction?
- Basis step (show true for n = 1), assumption (assume true for n = k), inductive step (show true for n = k + 1 using the assumption), and a full concluding sentence.
- Why do students lose marks on induction proofs?
- Most often for a missing or vague conclusion, a basis step without both sides shown, assuming what must be proved, or algebra errors in the inductive step.