Step by step
The question
Question: prove that if n is odd, then n² + n + 1 is odd.
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: n² + n + 1 becomes (2a + 1)² + (2a + 1) + 1.
3 Work it out
(2a + 1)² means the bracket times itself.
Every part times every part: 4a² + 2a + 2a + 1 + 2a + 1 + 1.
Like terms together: 4a² + 6a + 3.
4 Show the form
We take out 2 from all but the 1: 2(2a² + 3a + 1) + 1.
2a² + 3a + 1 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 n² + n + 1 is odd.
The square at the end means: the proof is done.
5 Check
An example: n = 3.
It gives 13.
13 is 2 times 6, 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.