00:01
Okay, so we've been given an inequality here and we've been asked to prove that it is true.
00:05
So the inequality we've been given is that 2 to the power of n is greater than n factorial.
00:17
And we need to prove that this is true.
00:19
And in this case, n is greater than or equal to 4.
00:23
Okay.
00:25
So when i was to prove this, i'm going to be proving by induction.
00:28
So to start off here, we're going to check the base case.
00:30
So the base case in this instance is going to be 4 because that's how lowest possible value of n or just the lowest and simplest possible value for n.
00:42
Yeah, so we'll be checking n equals 4 here.
00:45
If you're unsure about inequalities and stuff like that, you can check 5.
00:48
It doesn't actually.
00:51
Actually, it might matter.
00:52
Yeah, it does matter.
00:53
Do the base case.
00:54
You'll have to do the lowest possible integer.
00:57
So in this case, it is 4.
00:59
Okay and so we're checking that left -hand side here we've got two to the power of n which equals two to power of four which equals 16 okay and on the right -hand side we've got n factorial which is going to equal and four factorial which is 24 okay 24 is greater than 16 so it's true for and equals four our next part of proof by induction here is we're going to assume true for n equals k.
01:38
Okay.
01:40
So we're assuming true for n equals k here.
01:42
So what that means is that 2 to the power of k is greater than k factorial.
01:49
And we're just going to assume this.
01:50
And then we're going to use this in our proof for n equals k plus 1.
01:57
And that will allow us to prove by induction.
02:01
Okay.
02:04
So we have on our left -hand side here, we're going to try and prove for k -1 here.
02:11
So on our left -hand side, for k -plus -1, we have two to the power of k -plus -1.
02:23
Yeah, on our lesson time, we have two to the power of k -1.
02:25
I'm going to rearrange this.
02:28
I think i'll start with the right -hand side, actually, for now, just to see what we're going to rearrange into.
02:33
Although i think it's generally pretty similar.
02:38
So we start with the right hand side here.
02:40
So the right hand side is n factorial, so it's going to be k plus 1 factorial.
02:46
And we are going to be using our assumption here to rearrange our k first 1 factorial.
02:52
So k plus 1 factorial is essentially just 1 times by 2, times by 3, dot, dot, dot, times by k, and then also times by k plus 1.
03:05
So you'll see here that actually this bit is going to just be our k factorial.
03:10
So now we can use our k factorial in this.
03:14
And it will allow us to prove that this is true using our assumption.
03:20
So we just try trying to with these inductors try and get in our assumption.
03:28
I mean it's vital to get our assumptions to prove by induction.
03:31
So if you can, you're just looking for that way to get in.
03:34
In this case, we've got our k factorial from claim.
03:37
1 factorial.
03:39
And of course that's also going to be times by k plus 1.
03:41
So it's k factorial times by k plus 1.
03:43
That's what we've got on this side.
03:45
Okay.
03:46
And on this side, what i'm going to do is i'm going to try and get a 2 to the power of k here.
03:51
So again, just trying to get that assumption into it.
03:55
So this is going to be equal to 2 to the power of k times by 2.
04:00
Because you can imagine 2 to the power of k plus 1 is just the same as 2 to the power of k, but times by another two because we're just increasing the power by one, so we're timesing by the same number again.
04:12
There's two to parr okay, time by two.
04:17
There's some explanations to be done here.
04:19
I imagine for a lot of these types of questions, they're going to be quite similar.
04:22
It does take a little bit of just thinking about it, but if you were to try and to make these, we're trying to prove that the left -hand side is greater than the right -hand side...