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