Algebraic proof

An algebraic proof shows something is true for every number, not just the ones you try. Use letters for general numbers: 2n for even, 2n + 1 for odd, and n, n + 1 for consecutive integers.

Key facts

  • Even: 2n. Odd: 2n + 1. Consecutive: n, n + 1, n + 2.
  • Multiple of 3: 3n. Show something is a multiple of k by factorising out k.
  • Trying examples is not a proof, but one counterexample disproves a claim.

Worked examples

Example 1

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.
  2. Add them. n + (n + 1) + (n + 2) = 3n + 3
  3. Factorise. 3n + 3 = 3(n + 1)
  4. 3 times an integer is a multiple of 3.

Answer: n + (n + 1) + (n + 2) = 3(n + 1)

Example 2

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

  1. (n² + 6n + 9) − (n² + 2n + 1) = 4n + 8
  2. = 4(n + 2), a multiple of 4.

Answer: 4(n + 2)