00:01
For our proof, we will first suppose that n is odd and then prove that the expression is even.
00:08
And then we will suppose that n is even and prove that the expression is even.
00:15
So first, suppose n is odd.
00:22
Then we can write n as 2k plus 1, where k is an integer.
00:34
And now we'll be doing a lot of algebra.
00:41
So thus, we could have our expression, which is 9 n squared plus n.
00:50
If n is 2k plus 1, then it'll be 9 .2k plus 1 squared plus 2k plus 1.
01:00
Let's go ahead and expand out the squared binomial there.
01:05
So it would be 9 times 4k squared squared plus.
01:09
4k plus 1 and then plus 2k plus 1 and then let's go ahead and distribute through the 9 and we would get 36 k squared plus 36k plus 9 plus 2k plus 1 and now let's combine like terms and and we'll get 36k squared plus 18, or excuse me, 38, getting a little ahead of myself, 38k plus 10.
01:57
So that is it all simplified.
02:00
Now, in order to write something as an even number, that is 2 being multiplied by an integer.
02:07
So we can actually factor out a 2, since each one of those coefficients is even...