Prove if n is odd, then 5n + 1 is even

Step by step

The question

Question: prove that if n is odd, then 5n + 1 is even.

1 Write it with letters

n is odd, so n = 2a + 1, where a is a whole number.

That is what odd means.

Careful!

An example shows one number.

The letters show every number at once.

2 Put them in

Now put them in: 5n + 1 becomes 5(2a + 1) + 1.

3 Work it out

Open the brackets: 10a + 5 + 1.

Like terms together: 10a + 6.

4 Show the form

Every term has a 2 in it.

We take out 2: 2(5a + 3).

5a + 3 is a whole number, because whole numbers added and multiplied stay whole.

2 times a whole number: so 5n + 1 is even.

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

5 Check

An example: n = 3.

It gives 16.

16 is 2 times 8: 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.