Prove if n is even, then 3n + 5 is odd

Step by step

The question

Question: prove that if n is even, then 3n + 5 is odd.

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: 3n + 5 becomes 3(2a) + 5.

3 Work it out

Open the brackets: 6a + 5.

4 Show the form

We take out 2 from all but the 1: 2(3a + 2) + 1.

3a + 2 is a whole number, because whole numbers added and multiplied stay whole.

2 times a whole number, plus 1: that is the form of an odd number.

So 3n + 5 is odd.

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

5 Check

An example: n = 2.

It gives 11.

11 is 2 times 5, plus 1: odd, 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.