Show why a relationship holds

By UK GCSE Mathematics · independent study · myAudience

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Show why a relationship holds

Checking a few values can reveal a pattern or disprove a universal claim, but it cannot establish an identity for every input. Algebraic reasoning follows from general expressions and valid transformations.

A representation makes the domain explicit. Write an even integer as 2n, where n is an integer. The next even integer is 2n+2. Their sum is 4n+2=2(2n+1), which has a factor 2 and is therefore even. The factorisation explains all permitted cases at once.

A counterexample needs only one permitted case where a claimed universal statement fails. For example, n²+n+1 is not always prime: n=4 gives 21. Explain why the chosen value is within the claimed domain and why the result contradicts the assertion.

Worked example: Show that the sum of two odd integers is even.

Represent the odd integers as 2m+1 and 2n+1, with m, n integers.

Add to obtain 2m+2n+2.

Factorise 2(m+n+1).

Answer: The sum is even because it is twice an integer.

Check your understanding: Does checking n=1, 2, 3 prove a formula for all integers?

Answer: No.

Why: Those checks cover only three cases; a general argument or relevant proof is needed.