Proof, Knowledge Organiser

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

Worked Examples

Prove that the sum of any two odd numbers is always even. Solution Let the two odd numbers be (2a + 1) and (2b + 1) Sum = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1) This is 2 × (integer), so it is always even.