AlgebraTopic Summary

Knowledge Organiser: Algebraic Proof

Part of Proof · GCSE GCSE Mathematics revision

This topic summary covers Knowledge Organiser: Algebraic Proof within Proof for GCSE Mathematics. Revise Proof in Algebra for GCSE Mathematics with 14 exam-style questions and 12 flashcards. This topic shows up very often in GCSE exams, so students should be able to explain it clearly, not just recognise the term. It is section 4 of 4 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 4 of 4

Practice

14 questions

Recall

12 flashcards

Knowledge Organiser: Algebraic Proof

Key Terms

Proof: A logical argument showing a statement is ALWAYS true
Disprove / counterexample: A single example showing a statement is false
Even number: Can be written as 2n
Odd number: Can be written as 2n + 1 (or 2n − 1)
Consecutive integers: n, n+1, n+2, … (increasing by 1)
Consecutive even numbers: 2n, 2n+2; consecutive odd: 2n+1, 2n+3

Must-Know Facts

Even = 2n; Odd = 2n + 1 for some integer n
To prove: show algebraically it works for ALL values, not just examples
To disprove: find ONE counterexample
End a proof by stating what you've shown (e.g. "This is always even")
Even × anything = even; Odd × Odd = Odd; Even + Odd = Odd
Perfect square = (n)² = n²; always ≥ 0

Key Algebraic Identities for Proof

Even number: 2n
Odd number: 2n + 1
Consecutive integers: n, n+1
Consecutive even: 2n, 2n+2
Consecutive odd: 2n+1, 2n+3
Multiple of 3: 3n; square: n²

Key Formulas

Even: 2n; Odd: 2n + 1 (n is any integer)
Consecutive integers: n, n+1, n+2
Consecutive even integers: 2n, 2n+2, 2n+4
Sum of 3 consecutive integers: 3n + 3 = 3(n+1) — always a multiple of 3

Common Mistakes

Using specific numbers: Showing it works for n = 5 is NOT a proof — must use general algebraic expressions
Even number represented as n+2: Use 2n for any even number — n+2 could be odd
Not fully expanding: Expand ALL brackets before simplifying to reveal the structure
Disproof: To disprove a statement, find ONE counterexample — you do not need a full algebraic argument

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Keep building this topic

Read this section alongside the surrounding pages in Proof. That gives you the full topic sequence instead of a single isolated revision point.

Worked Example Proving It's ALWAYS True Key Representations

Practice Questions for Proof

Which expression represents an even number for all integer values of n?

A. n + 2
B. 2n
C. 2n + 1
D. n²

1 markfoundation

Prove by exhaustion that when you divide any single-digit positive integer by 4, the remainder is always 0, 1, 2, or 3. Show your working by testing all possible cases.

2 markshigher

Quick Recall Flashcards

What are the three steps in an algebraic proof?

1. State what your letters represent (e.g. let n be any integer) 2. Manipulate algebraically (expand, simplify, factorise) 3. Show the result matches the required form with a clear conclusion

What is a counter-example in mathematics?

A single specific example that shows a statement is NOT always true. One counter-example is enough to disprove a conjecture.

14 questions on Proof — practise free

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Worked Example

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Proof