Proof questions in A-Level Maths: every type and how to answer it

Proof is one of the smallest topics on the specification by content, and one of the most consistently mishandled in the exam hall. Students lose marks not because they can't do the algebra, but because they don't know what a complete, rigorous argument is supposed to look like on paper. A correct final line with gaps in the logic before it scores badly, and a slightly clumsy argument that is logically watertight scores well.

This guide covers every proof type on the Edexcel, AQA, OCR and OCR MEI specifications: proof by deduction, proof by exhaustion, proof by contradiction, and disproof by counter-example. Each has its own structure and its own way of losing easy marks.

What examiners actually want from a proof

A proof is a chain of statements, each one following logically from the last, that starts from something known (a definition, an assumption, or given information) and ends at the statement you were asked to prove. Every link in that chain has to be justified — you cannot skip a step because it 'obviously' follows.

• State what you are assuming or starting from, explicitly.
• Show every algebraic step; do not jump from line 2 to line 5.
• Use correct mathematical language — 'let', 'suppose', 'then', 'therefore', 'hence' — rather than ordinary speech.
• End with a clear concluding statement that refers back to what was being proved, not just a final equation.
• Define your variables. If n is an integer, say so before you use it.

1. Proof by deduction

This is the most common type at both GCSE and A-Level. You start from a definition or a given statement and manipulate it algebraically until you reach the result. Most 'prove that' questions on identities, inequalities and number properties are proof by deduction.

Worked example: an algebraic identity

Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all positive integers n.

1. Expand: (2n + 1)² − (2n − 1)² = (4n² + 4n + 1) − (4n² − 4n + 1).
2. Simplify: = 8n.
3. State the conclusion: since n is an integer, 8n is a multiple of 8 for all positive integers n. Hence proved.

Notice the last line does two jobs: it restates the algebra in the language of the question ('multiple of 8'), and it explicitly says the proof holds 'for all positive integers n' because that was the claim. A bare '= 8n' with no final sentence typically loses the last mark even though all the maths is correct.

Worked example: odd and even numbers

Prove that the sum of two consecutive odd numbers is always a multiple of 4.

1. Let the first odd number be 2n + 1, where n is an integer.
2. The next consecutive odd number is 2n + 3.
3. Sum: (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1).
4. Since n is an integer, n + 1 is an integer, so 4(n + 1) is a multiple of 4. Hence proved.

2. Proof by exhaustion

Proof by exhaustion works by checking every possible case, when the number of cases is small and finite. It appears at A-Level typically for statements about a limited set of integers, or about remainders when dividing by a small number.

Example: prove that n² + n is even for all integers n. Since every integer is either even or odd, there are exactly two cases to check.

Both cases give an even result, and since every integer falls into one of these two cases, the statement is true for all integers. The key phrase examiners look for is a statement that the cases considered are exhaustive — that there is no third possibility being missed.

3. Proof by contradiction

This is usually the hardest type for students to structure, and it appears on both Edexcel and AQA A-Level Pure papers, most famously for proving that √2 is irrational and that there are infinitely many primes.

1. Assume the opposite of what you want to prove is true.
2. Follow logical steps from that assumption.
3. Reach a contradiction — a statement that cannot possibly be true, or that contradicts the assumption itself.
4. Conclude that the original assumption must be false, so the statement you wanted to prove is true.

Worked example: √2 is irrational

Assume, for contradiction, that √2 is rational. Then √2 can be written as a/b, where a and b are integers with no common factors (the fraction is in its simplest form) and b ≠ 0.

1. Squaring: 2 = a²/b², so a² = 2b².
2. This means a² is even, so a must be even (since an odd number squared is always odd). Write a = 2c.
3. Substitute: (2c)² = 2b², so 4c² = 2b², so b² = 2c².
4. This means b² is even, so b must also be even.
5. But a and b were assumed to have no common factors, and we've now shown both are even — a contradiction.
6. Therefore the original assumption is false, and √2 is irrational.

Worked example: infinitely many primes

Assume, for contradiction, that there are finitely many primes: p₁, p₂, ..., pₙ. Let N = p₁ × p₂ × ... × pₙ + 1. N is either prime itself, or has a prime factor. But dividing N by any of p₁ through pₙ leaves remainder 1, so none of them divide N. So N's prime factor (or N itself) is a prime not in the original list — a contradiction. Therefore there are infinitely many primes.

4. Disproof by counter-example

This is the fastest proof type to write and the one students most often overcomplicate. To disprove a general statement, you only need one example where it fails — you do not need to explain why it fails in general.

Example: 'n² + n + 41 is prime for all positive integers n' — disprove this. Testing small values works for a long time (it is prime for n = 1 up to n = 39), which is exactly why the question is set: it tests whether you'll actually check rather than assume the pattern continues. At n = 40: 40² + 40 + 41 = 1681 = 41², which is not prime. One correctly evaluated counter-example is a complete disproof.

Common mistakes across all proof types

How to revise proof effectively

• Learn the four proof types by name and structure before attempting mixed practice — you can't choose the right method if you don't recognise which type a question needs.
• Practise writing the concluding sentence separately; it is the most commonly dropped mark.
• Redo the standard proofs (√2 irrational, infinitely many primes) from memory until the structure is automatic.
• When practising deduction proofs, write every algebraic line out in full rather than combining steps in your head — skipped lines are where marks and errors both hide.

Proof rewards precision of language as much as correct algebra. Once you know which of the four structures a question needs and you write the concluding sentence every time, this becomes one of the more reliable topics on the paper rather than one of the most feared.

A worked FAQ on proof

Do I need to write 'QED' or a box at the end?

No UK exam board requires a QED symbol or a box. What they do require is an unambiguous concluding sentence that refers back to the original claim — phrases like 'as required', 'hence proved' or 'therefore the statement is true for all integers n' do the job. A bare tick or a box with no words attached is not credited as the concluding statement.

Can I use a specific number to prove a general statement?

Only for disproof by counter-example. For every other proof type, testing one or two numbers is useful for checking your own understanding before you write your answer, but it is never sufficient on its own to prove a statement true for 'all' integers, or 'all' odd numbers, or similar. Examiners specifically design mark schemes to award zero for 'I tried n = 1, 2, 3 and it worked' when the question says 'prove for all n'.

What is the difference between proof by exhaustion and just trying examples?

The difference is completeness. Exhaustion is only valid when you can list every possible case and there is no case left unchecked — for example, 'n is even' or 'n is odd' genuinely covers every integer. Trying examples without covering every case (say, checking n = 1 to 10 and stopping) is not a proof at all, because there could always be a counter-example just beyond where you stopped.

Proof by induction: a brief note

Proof by induction is not on the standard A-Level Maths specification, but it appears on A-Level Further Maths (Edexcel, AQA and OCR all include it). It proves a statement is true for all positive integers n by showing it holds for a base case (usually n = 1), then showing that if it holds for n = k, it must also hold for n = k + 1. If you are taking Further Maths, treat induction as a distinct fifth proof type with its own four-part structure: basis, assumption, inductive step, and conclusion — each part is separately credited on the mark scheme, and missing the concluding statement here loses marks just as often as it does in the four types covered above.

A quick self-test before the exam

Recognising the phrasing before you start writing saves time and prevents you from setting up the wrong structure halfway through an answer, which is a difficult position to recover from under exam conditions.