00:01
Hello, so here we're looking to show that if we add powers of five, so one plus five, plus 25, and so on, up to plus five to the n is, well, always going to be equal to five to the n plus one minus one, all divided by four.
00:23
And this is going to be true for all n, so for all n greater than equal to zero.
00:29
Okay, so to prove this by induction, well, the basis case is just when, so we're going to prove that p of n is always true.
00:37
What we look at this p of zero, the initial case, we look when if n is, we get zero plus one is just going to be one.
00:46
So we get five to the one minus one over four.
00:50
So the initial case is just five to the one minus one over four.
01:00
That's is 5 to the 1 is 5 minus 1 is 4 over 4.
01:05
So it's going to be holding true when n is equal to 0.
01:09
Now, for the inductive step, we say that p of n holds true.
01:13
Let's say n being equal to k, and they want to show that k plus 1 is true...