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Algebra · Higher

Algebraic proof

Algebraic proof uses algebra to prove that a statement is always true (or always false). You show that an expression is equivalent to the required form for all values of the variable.

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Revising algebraic proof the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Algebraic proof GCSE” on the site for the complete guide.

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Key facts to remember

  • 1An even number can be written as 2n; an odd number as 2n + 1 (where n is an integer).
  • 2Consecutive integers: n, n+1, n+2; consecutive even: 2n, 2n+2; consecutive odd: 2n+1, 2n+3.
  • 3To prove an expression is always even, show it equals 2 × (integer).
  • 4To prove an expression is always odd, show it equals 2 × (integer) + 1.
  • 5Expand and simplify fully, then factorise to show the required property.
  • 6A counterexample (one value that fails) is enough to disprove a statement.
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Worked examples

Example 1

Prove that the sum of three consecutive integers is always a multiple of 3.

Working

  1. Let the integers be n, n + 1, n + 2
  2. Sum = n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
  3. 3(n + 1) is always a multiple of 3
AnswerProven: the sum = 3(n + 1) which is always divisible by 3.
Example 2

Prove that (n + 3)² − (n + 1)² is always a multiple of 4.

Working

  1. Expand (n + 3)² = n² + 6n + 9
  2. Expand (n + 1)² = n² + 2n + 1
  3. Difference = (n² + 6n + 9) − (n² + 2n + 1) = 4n + 8 = 4(n + 2)
  4. 4(n + 2) is always divisible by 4
AnswerProven: the expression = 4(n + 2), always a multiple of 4.
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Common mistakes

✗Using specific numbers (e.g. n = 3) — this shows an example, not a proof.
✗Not fully expanding brackets before simplifying.
✗Failing to factorise the result to show divisibility.
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Exam tips

✓Always use algebraic expressions (2n, 2n+1 etc.) — never use specific numbers in a proof.
✓End by clearly stating what you have shown, e.g. "This is divisible by 4 for all integer values of n."

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