An algebraic proof shows that a statement is true for every number, not just the ones you test. Represent numbers with algebra: 2n is even, 2n + 1 is odd, and n, n + 1, n + 2 are consecutive integers. Expand and simplify, then factorise to show the result: for example, an expression is a multiple of 8 if it can be written as 8 × (an integer).
Building blocks for proof
- Even: 2n. Odd: 2n + 1
- Consecutive integers: n, n + 1, n + 2, …
- Consecutive even numbers: 2n, 2n + 2
- Multiple of k: k × (an integer)
- n(n + 1) is always even
Why examples are not proof
Checking a statement for a few numbers shows it might be true, but not that it is always true. Algebra uses a letter to stand for any integer, so a single argument covers every case. One counterexample, however, is enough to show a statement is false.
Representing numbers
Choose expressions that capture the numbers in the statement. Any even number is 2n and any odd number is 2n + 1. Consecutive integers are n, n + 1 and n + 2. If two numbers can be different, use different letters, such as 2a + 1 and 2b + 1.
Proof by expanding
Many proofs come down to expanding brackets and simplifying, after which the result appears. To show (n + 5)² − (n + 1)² is a multiple of 8, expand: n² + 10n + 25 − (n² + 2n + 1) = 8n + 24. Then factorise: 8(n + 3), which is 8 times an integer.
Proving multiples, odd and even
To prove something is a multiple of k, factorise it so that k is outside the bracket and the bracket is an integer. To prove a number is even, write it as 2 × (an integer); to prove it is odd, write it as 2 × (an integer) + 1. A useful fact: n(n + 1) is always even, because one of two consecutive integers must be even.
Writing the conclusion
Finish every proof with a sentence that links the algebra to the statement, such as “8(n + 3) is a multiple of 8, so the expression is always a multiple of 8”. Without this sentence, the final mark is often lost.
Worked examples
Example 1. Prove that the sum of four consecutive integers is always even.
- n + (n + 1) + (n + 2) + (n + 3) = 4n + 6.
- 4n + 6 = 2(2n + 3).
Answer: 2(2n + 3) is a multiple of 2, so the sum is always even.
Example 2. Show that (n + 5)² − (n + 1)² is always a multiple of 8.
- Expand: n² + 10n + 25 − n² − 2n − 1 = 8n + 24.
- Factorise: 8(n + 3).
Answer: 8(n + 3) is a multiple of 8 for every integer n.
Example 3. Prove that (3n + 1)² − (3n − 1)² is a multiple of 12.
- (9n² + 6n + 1) − (9n² − 6n + 1) = 12n.
Answer: 12n is 12 × n, so it is always a multiple of 12.
Example 4. Prove that the product of two consecutive even numbers is a multiple of 8.
- Let the numbers be 2n and 2n + 2: 2n(2n + 2) = 4n(n + 1).
- n(n + 1) is a product of consecutive integers, so it is even: n(n + 1) = 2k.
- So 4n(n + 1) = 8k.
Answer: The product is 8k, a multiple of 8.
Example 5. Show that “n² − n + 11 is always prime” is false.
- Try n = 11: 121 − 11 + 11 = 121.
- 121 = 11 × 11.
Answer: n = 11 is a counterexample.
Common mistakes
- Using examples as a proof. Examples can disprove a statement but cannot prove one.
- Using the same letter for different numbers. Two different odd numbers need 2a + 1 and 2b + 1.
- Errors when subtracting a bracket. −(n + 1)² = −n² − 2n − 1: change every sign.
- Stopping before factorising. 8n + 24 must be written as 8(n + 3) to show it is a multiple of 8.
- Missing the concluding sentence. State clearly what the algebra shows.
How algebraic proof is examined
Algebraic proof is a Higher tier topic in Edexcel GCSE (1MA1), and the 4MA1 International GCSE includes similar “show that” questions. Typical questions:
- prove a result about odd, even or consecutive integers
- expand and simplify an expression, then show it is a multiple of a number
- prove an identity, such as showing one expression equals another
- find a counterexample to disprove a statement
Practise with the algebraic proof worksheet, which has full worked solutions, and the GCSE algebra revision engine.
Proof relies on expanding brackets and factorising. For proofs with primes, see primes and proof.
Frequently asked questions
How do you write a general even number?
As 2n, where n is any integer.
How do you prove something is a multiple of 5?
Write it as 5 × (an integer), usually by factorising out a 5.
Is checking lots of examples a proof?
No. Only algebra that works for every value of n proves a statement.