Proof
Algebraic proof is the process of showing a mathematical statement is true for every case (or false, using a single counter-example) by writing numbers algebraically and manipulating expressions logically. GCSE Higher papers commonly ask you to prove results about consecutive integers, odd and even numbers, or geometric facts, using expressions such as n, n + 1 or 2n + 1.
Before you start
Make sure you're comfortable with these topics first:
Method
- Represent the numbers involved algebraically, for example consecutive integers as n, n+1, n+2, even numbers as 2n, and odd numbers as 2n + 1.
- Write an expression for the statement you are proving, such as a sum, difference or product of these algebraic terms.
- Expand any brackets and simplify the expression fully, collecting like terms.
- Factorise the simplified expression to reveal a common factor that proves the required property, such as factoring out 3 to show a multiple of 3.
- Write a concluding sentence linking the algebra back to the original statement, explaining why the factorised form proves the claim.
- To disprove a statement, find just one counter-example that fails it and show the working clearly.
Worked example
Prove that the sum of three consecutive even numbers is always a multiple of 6.
- Write the three consecutive even numbers algebraically as 2n, 2n + 2 and 2n + 4.
- Add them together: 2n + (2n + 2) + (2n + 4).
- Simplify: 6n + 6.
- Factorise: 6n + 6 = 6(n + 1).
- Since 6(n + 1) is 6 multiplied by an integer, it is always a multiple of 6.
- Final answer: 2n + (2n + 2) + (2n + 4) = 6n + 6 = 6(n + 1), a multiple of 6 for any integer n.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Prove algebraically that the sum of five consecutive integers is always a multiple of 5.
Prove algebraically that (3n + 1)^2 - (3n - 1)^2 is always a multiple of 12, for any positive integer n.
OAB is a triangle, where OA = vector a and OB = vector b. P is the point on OA such that OP = (2/3)OA, and Q is the point on OB such that OQ = (2/3)OB. Prove that PQ is parallel to AB, and find PQ in terms of a and b.
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