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IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Algebra

Algebraic Proof

A proof shows something is true for every number, not just the ones you tried. Edexcel Unit 2 proofs are usually 3–4 marks. They reward a clear algebraic setup, correct expansion, and a final sentence that links the algebra to the statement.

1Writing numbers algebraically

Every proof starts by writing the numbers in the statement using a letter. Let n be any integer.

NumberAlgebra
Any even number2n
Any odd number2n + 1
Consecutive integersn, n + 1, n + 2
Consecutive even numbers2n, 2n + 2
Consecutive odd numbers2n + 1, 2n + 3
A multiple of 55n
Two different odd numbers2n + 1 and 2m + 1
Key ideaTo show something is a multiple of k, write it as k × (an integer). To show it is odd, write it as 2 × (an integer) + 1.
Common trap
  • Using 2n + 1 for both of two different odd numbers. That only proves the result when the two numbers are equal. Use a second letter (2m + 1).

2Proving a statement about integers

Worked example
Prove that the sum of any three consecutive integers is a multiple of 3.
  1. Let the integers be n, n + 1 and n + 2, where n is an integer.
  2. Sum = n + (n + 1) + (n + 2) = 3n + 3.
  3. Factorise: 3n + 3 = 3(n + 1).
  4. n + 1 is an integer, so 3(n + 1) is a multiple of 3.
The sum is a multiple of 3
Exam tipThe last line earns a mark on its own. Always finish with a sentence that explains why your expression proves the statement.

3Proofs that need expanding

Worked example
Prove that the difference between the squares of two consecutive odd numbers is always a multiple of 8.
  1. Let the odd numbers be 2n + 1 and 2n + 3.
  2. (2n + 3)² − (2n + 1)² = (4n² + 12n + 9) − (4n² + 4n + 1).
  3. = 8n + 8 = 8(n + 1).
  4. n + 1 is an integer, so the difference is a multiple of 8.
The difference is a multiple of 8
Worked example
Prove that the square of any odd number is odd.
  1. Let the odd number be 2n + 1.
  2. (2n + 1)² = 4n² + 4n + 1 = 2(2n² + 2n) + 1.
  3. 2n² + 2n is an integer, so the square is of the form 2k + 1, which is odd.
The square of an odd number is odd

4Proving identities (≡)

To prove an identity, start with one side (usually the more complicated one) and manipulate it until it becomes the other side. Do not move terms across the ≡ sign.

Worked example
Prove that (x + 2)² − (x − 2)² ≡ 8x.
  1. LHS = (x² + 4x + 4) − (x² − 4x + 4).
  2. = x² + 4x + 4 − x² + 4x − 4.
  3. = 8x = RHS.
LHS ≡ RHS

5Proofs using completing the square

A squared term is never negative. So if you can write an expression as (something)² + a positive number, it is always positive.

Worked example
Prove that x² − 6x + 11 > 0 for all values of x.
  1. Complete the square: x² − 6x + 11 = (x − 3)² − 9 + 11 = (x − 3)² + 2.
  2. (x − 3)² ≥ 0 for all x.
  3. So (x − 3)² + 2 ≥ 2 > 0 for all x.
x² − 6x + 11 > 0 for all x
xy-1123456724681012min (3, 2)
The minimum point is (3, 2), above the x-axis, so the graph never goes below zero.

6Disproving: counterexamples

To show a statement is false, you only need one example where it fails. This is called a counterexample.

Worked example
“n² + n + 41 is prime for every positive integer n.” Show this is false.
  1. Try n = 41: 41² + 41 + 41 = 41(41 + 1 + 1) = 41 × 43.
  2. This has factors 41 and 43, so it is not prime.
n = 41 is a counterexample
Key ideaExamples can never prove a statement is always true. However many cases you check, you must use algebra for a proof. But one counterexample is enough to disprove it.
Common mistakes
  • Checking a few numbers and calling it a proof.
  • Using the same letter for two different numbers (n and n for two different odd numbers).
  • Expanding brackets incorrectly, especially subtracting a bracket: −(4n² + 4n + 1) = −4n² − 4n − 1.
  • Missing the concluding sentence, e.g. “… so it is a multiple of 8”.
  • In an identity proof, working on both sides at once or moving terms across ≡.

★Key Vocabulary Recap

Proof
An argument that shows a statement is true in all cases.
Integer
A whole number: …, −2, −1, 0, 1, 2, …
Identity (≡)
An equation true for every value of the variable.
Counterexample
A single example that shows a statement is false.
Consecutive
Following one after another, e.g. 7, 8, 9.
Specification reference (Pearson Edexcel 4XMA1, Unit 2 Higher)
  • Use algebra to support and construct proofs.
  • Prove identities; use completing the square in proofs.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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