00:01
Okay, we'll first suppose that n is an even number, and then we'll suppose that n is an odd number and prove that in both cases the result is even.
00:11
So let's first suppose that n is even, n is even.
00:25
Then we can write n as a multiple of 2, which would be 2k, where k is an integer.
00:37
And now we'll go ahead and replace n in the expression with 2k and see if it comes out to be an even integer.
00:50
So now, n squared minus 3n plus 2.
00:58
Well, that is now equal to 2k squared minus 3 times 2k plus 2.
01:06
So notice how i've just replaced the n with 2k.
01:09
Let's go ahead and simplify, and we'll get 4k squared minus 6k plus 2.
01:17
And now we want to factor out a 2 from each of those terms, and we'll get 2 times 2k squared minus 3k plus 1.
01:31
So now 2k squared minus 3k plus 1 is an end.
01:40
Integer, so 2 times 2k squared minus 3k plus 1 is even.
01:51
So we're good there...