Proof – GCSE Maths

Higher · Grade 8 · Non-calculator

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Algebraic proof at GCSE is identities and 'show that $n$ is always…'. Even numbers $2k$, odd $2k+1$, consecutive $n,n+1,n+2$, consecutive even $2n,2n+2$. Expand, collect, and factor to the form the question named ('a multiple of $4$', 'always odd'). Every line needs an equals sign that is actually true, not a wish.

Counter-examples disprove a false claim: one number that fails is enough. For 'show that $(n+1{)}^2-(n-1{)}^2=4n$', expand both squares and subtract — do not start from $4n$. Finish with 'which is a multiple of $4$' or 'as required'.

To prove a statement for all integers, write even as $2n$ and odd as $2n+1$ (use a second letter if there are two numbers). Expand, collect, and factor out 2 (even) or write $2(\dots )+1$ (odd). A single counter-example kills a 'always' claim.

Objectives

• Write a general integer with a letter.
• Expand and factor the expression.
• Do not test one number and stop.

Key points

• Represent numbers algebraically: any integer $n$, an even number $2n$, an odd number $2n+1$, consecutive integers $n,n+1,n+2$.
• Use different letters ($n$ and $m$) for two numbers that are not linked.
• Expand brackets, including $(2n+1{)}^2$, and simplify accurately.
• Factorise the result to show it is a multiple of the required number, or write it as $2(\dots )+1$ to show it is odd.

Worked examples

Questions and answers

GCSE Maths proof questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Testing a few numbers and calling it a proof

   Use a letter such as n so the argument covers every integer

2. Writing two consecutive odd numbers as 2n+1 and 2n+1

   Consecutive odd numbers are 2k+1 and 2k+3

3. Stopping at 4k+4

   Factor to 4(k+1) and say that k+1 is an integer

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Start with a 'Let …' line that defines each letter. It is often the first mark and it stops you mixing letters up.
• Show the factorised form with the required factor at the front, then state in words why that proves the result.

Printable worksheet

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Questions students ask

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More on Proof

• Proof worked examples and method
  The method in order, key facts and the full lesson explanation

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.