Prove if n is even, then n^2 is even

Step by step

The question

Question: prove that if n is even, then n² is even.

1 Write it with letters

n is even, so n = 2a, where a is a whole number.

That is what even means.

Careful!

An example shows one number.

The letters show every number at once.

2 Put them in

Now put them in: n² becomes (2a)².

3 Work it out

(2a)² means the bracket times itself.

Every part times every part: 4a².

4 Show the form

Every term has a 2 in it.

We take out 2: 2 × 2a².

2a² is a whole number, because whole numbers added and multiplied stay whole.

2 times a whole number: so n² is even.

The square at the end means: the proof is done.

5 Check

An example: n = 2.

It gives 4.

4 is 2 times 2: even, as the proof says.

Right!

Let's look back.

First: write it with letters.

Then: put them in.

Then: work it out.

Then: show the form.

Last: check.