00:02
Hi, here we have to prove the following statement that n to the q plus 5n is a multiple of three for any integer n and we're positive integer n and we want to prove this using mathematical induction.
00:16
So first of all, we are going to need this formula.
00:20
This is the binomial formula for n equals 3.
00:24
So we have a plus b to the cube is a to the cube plus 3, a square b.
00:30
Plus 3ab square plus b to the cube.
00:33
We are going to use this formula.
00:37
So the proposition, the statement we want to show is this, that n to the cube plus 5n is equal 3 times s for some s, for some positive integer s and this is the statement we are going to call this statement let me just make it a bit clear so i'm going to make these statements okay so just bear with me one sec to make this thing clear so what we have is and to the cube plus 5n is 3 times s for some positive integer s and i'm going to call this statement p of n now this means what i have here this equation that n to the cube plus 5n is a multiple of 3 a number is a multiple of 3 if it is 3 times another number since we are living in the set of positive integers and this is a positive integer if n is a positive integer or all of this thing is a positive integer.
02:08
So everything has to be positive.
02:10
So it's going to be three times s.
02:12
So for some other positive integer s.
02:16
Okay.
02:17
So and we want to show that this is true for any positive integer n.
02:21
That means we want to show it that it is true for all n that greater equal to one.
02:28
So our n can be one, two, three, four, et cetera.
02:32
Okay.
02:33
So using mathematical...