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Algebraic Proof

HigherHigher tier onlyAQAEdexcelOCR

Master algebraic proof for GCSE Maths with structured, exam-style practice. This Higher resource covers constructing algebraic proofs and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Use 2n for an even number and 2n+1 for an odd number.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Algebraic proof asks you to show that a statement is always true, not merely true for the numbers you happened to test. Checking examples is never a proof, however many you try.

The technique is to represent the general case in algebra. Any integer is \(n\); consecutive integers are \(n\), \(n+1\) and \(n+2\); an even number is \(2n\) and an odd number is \(2n + 1\).

You then manipulate the expression and show it has the required property. Ending with a factorised form is usually the goal, since \(2(\ldots)\) proves a result is even and \(3(\ldots)\) proves it is a multiple of three.

Revision notes

Representing numbers algebraically

Use \(2n\) for even and \(2n + 1\) for odd, where \(n\) is any integer. Consecutive numbers follow on as \(n+1\), \(n+2\).

Using separate letters for two different unknown numbers matters — writing \(2n\) twice would force them to be equal.

Manipulating and factorising

Expand and simplify, then factorise to reveal the property required.

To show the sum of two odd numbers is even: \((2m+1) + (2n+1) = 2m + 2n + 2 = 2(m+n+1)\). The factor of \(2\) proves it is even.

Writing the conclusion

State explicitly why the factorised form proves the claim.

A final line such as since the result is 2 times an integer, it is even is required. Stopping at the algebra without the conclusion loses the final mark.

Key points

  • Testing examples is not a proof.
  • Use \(2n\) for even and \(2n+1\) for odd.
  • Use different letters for different unknowns.
  • Consecutive integers are \(n\), \(n+1\), \(n+2\).
  • Factorise to reveal the required property.
  • State a clear conclusion at the end.

Worked examples

Example 1

Prove that the sum of two consecutive integers is odd.

Working

\[n + (n + 1)\]represent two consecutive integers
\[= 2n + 1\]simplify the expression
\[\text{This is odd by definition}\]2n + 1 is the general form of an odd number

Example 2

Prove that the sum of three consecutive integers is a multiple of \(3\).

Working

\[n + (n+1) + (n+2)\]represent three consecutive integers
\[= 3n + 3\]simplify
\[= 3(n + 1) \text{, a multiple of 3}\]factorise to reveal the factor of 3

Example 3

Prove that the product of two even numbers is a multiple of \(4\).

Working

\[2m \times 2n\]represent two even numbers using different letters
\[= 4mn\]multiply the expressions
\[\text{4 times an integer, so a multiple of 4}\]state the conclusion

Common mistakes

  • Testing numbers instead of proving.

    Showing it works for 3 and 5 proves nothing about every case.

  • Using the same letter for two different numbers.

    2n and 2n are the same even number. Use 2m and 2n for two different ones.

  • Stopping at the algebra.

    The conclusion explaining why the result follows is worth a mark.

  • Factorising incorrectly.

    3n + 3 factorises to 3(n + 1), not 3(n + 3).

Exam tips

  • Define your letters at the start, stating they are integers.
  • Use different letters for different unknown numbers.
  • Aim to finish with a factorised expression.
  • Write a concluding sentence explaining what the factorisation shows.

Key terms

Proof
A logical argument showing a statement is always true.
Integer
A whole number, positive, negative or zero.
Consecutive
Following one after another.
Multiple
The result of multiplying by a whole number.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.