The idea

Expressing generality — the language of proof

Why testing examples proves nothing, how 2n, 2n + 1 and n, n + 1, n + 2 encode "any even", "any odd" and "consecutive", and when one counterexample settles everything.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Algebra · Algebraic proof

Expressing generality — the language of proof

Why testing examples proves nothing, how 2n, 2n + 1 and n, n + 1, n + 2 encode "any even", "any odd" and "consecutive", and when one counterexample settles everything.

Why it works

Evidence is not proof

"Show this works for 3, for 7, for 100" is evidence, not proof — the claim is about EVERY number, and no pile of examples covers them all. Algebra's letters do what examples can't:

n

stands for any integer at once, so one calculation with

n

is every calculation simultaneously.

The vocabulary

The standard encodings, with

n

any integer:

any even number: $2n$ (a multiple of 2, by construction);
any odd number: $2n + 1$ (one more than an even);
consecutive integers: $n, \; n + 1, \; n + 2, \ldots$ ;
consecutive EVEN numbers: $2n, \; 2n + 2, \; 2n + 4$ (they step by 2);
consecutive ODD numbers: $2n + 1, \; 2n + 3$ ;
any multiple of $k$ : $kn$ .

Keep reading — free

The rest of the explanation, plus 2 worked examples you step through move by move.

Start free

Takes a minute — no card.