Prove if n^2 + 2n is odd, then n is odd Step by Step

Step by step

The question

Question: prove that if n² + 2n is odd, then n is odd.

1 Turn it around

An if-then statement means the same as its contrapositive: if not Q, then not P.

We saw it with truth tables.

if P, then Q ≡ if not Q, then not P

Turned around, it says: if n is even, then n² + 2n is even.

Starting from n is much easier: we know how to write it with letters.

turned around: if n is even, then n² + 2n is even

Careful!

That swaps the two parts.

It is the converse, a different statement.

Turned around means: swap the parts and put not on both.

Careful: not if n is odd, then n² + 2n is odd ✗

2 Write it with letters

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

n = 2a

That is what even means.

even = 2 × a whole number

3 Put them in

Now put them in: n² + 2n becomes (2a)² + 2(2a).

n² + 2n = (2a)² + 2(2a)

4 Work it out

(2a)² means the bracket times itself.

= (2a)(2a) + 2(2a)

Every part times every part: 4a² + 4a.

= 4a² + 4a

5 Show the form

Every term has a 2 in it.

We take out 2: 2(2a² + 2a).

= 2(2a² + 2a)

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

2a² + 2a is a whole number

2 times a whole number: so n² + 2n is even.

So n² + 2n is even.

The turned statement is true, so the first one is true too: if n² + 2n is odd, then n is odd.

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

So: if n² + 2n is odd, then n is odd. ∎

6 Check

Try n = 1.

Then n² + 2n is 3: odd.

n = 1: n² + 2n = 3, odd

And 1 is odd, as the statement says.

Right!

and 1 is odd

Let's look back.

First: turn it around.

Then: write it with letters.

Then: put them in.

Then: work it out.

Then: show the form.

Last: check.

Answer