Simple deductive proof

Maths: Analysis and Approaches · Number and algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A deductive proof shows that a statement is true in every case, using steps that each follow from the one before. Checking a few numbers is not a proof, because it leaves the other cases untested.

To prove an identity, start with one side (normally the left-hand side), change it step by step with correct algebra, and finish when it matches the other side. Do not move terms across as if you were solving an equation.

An equation (=) is true only for some values of x. An identity (≡) is true for all values. For proofs about whole numbers, write the numbers in algebra first, for example 2n for an even number.

Rules to remember

= is true for some values (3x + 2 = 11 only when x = 3); ≡ is true for all values (3(x + 2) ≡ 3x + 6)
Even number: 2n. Odd number: 2n + 1 or 2n − 1, where n is an integer
Consecutive integers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2
A multiple of k is written k × (an integer); factorise to show it
One of any two consecutive integers is even, so n(n + 1) is always even
Layout: LHS ≡ ... ≡ ... ≡ RHS, one working step per line, then a concluding sentence

Common mistake

Starting from the statement you are asked to prove and working on both sides at once. That assumes the result is true; begin with one side only and work until you reach the other.

Worked example

Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8.

Let the odd numbers be 2n − 1 and 2n + 1, where n is an integer.
(2n + 1)² − (2n − 1)² ≡ (4n² + 4n + 1) − (4n² − 4n + 1).
≡ 4n + 4n ≡ 8n.
n is an integer, so 8n is 8 × (an integer).

Answer: The difference is 8n, which is always a multiple of 8.

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Topic 1: Number and Algebra

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