00:01
Okay, so we want to use mathematical induction to show that the following is true.
00:05
So let's start with condition 1.
00:07
So let's check if n is equal to 1 is true.
00:10
So we get 1 cubed plus 2 times 1, so that's equal to 3.
00:14
And 3 is divisible by 3.
00:17
So this is true.
00:19
And now let's go on to condition 2.
00:21
In condition 2, we assume that k plus 1 is true.
00:27
So that is n, or actually k cubed, plus 2k.
00:34
Is divisible by 3.
00:37
And now let's show that n is equal to k plus 1 is divisible by 3 as well.
00:42
So let's write that out.
00:44
We get k plus 1 to the power of 3 plus a 2 times k plus 1.
00:49
Let's note that we can factor out a k plus 1 since we have that in common.
00:53
And we're left with a k plus 1 to the power of 2 and then plus 2.
00:59
So multiplying that out, we get k squared plus 2k...