In Exercises 25–34, use mathematical induction to prove that each...

Question

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

2 is a factor of n^2 - n.

Accepted Answer

Hello. Today we're going to be using mathematical induction to prove that three is a factor of N cubed plus two N. Now what it means to show that three is a factor of N. Q plus two. N is to show that N cubed plus two. N is divisible by three. So let's go ahead and start with the first step of the mathematical induction. And the first step is to show that in Q plus two, N is at least equal to three. And we're going to do that by allowing end to equal to one. So when N is equal to one we get the statement three is equal to end cubed which is one cubed plus two N which is two times +11 cubed is going to give us one and two times one is going to give us too. So we have three is equal to one plus two and one plus two is equal to three which at least proves the first statement that N Q plus two. N can equal to three. Now we want to go ahead and allow N two equal to K. So the second step is to allow N two equal to K. And this is going to give us the statement the function P K which is equal to K cubed plus two K. And that is going to equal to three. K. I'm sorry not three K. Three are now the reason why we introduced this our variable is because we're going to let our represent any positive integer when r is any positive integer multiplied by three. That means that that value is always going to be divisible by three. So we're going to use this statement in the next step to help us prove that this statement is true. And the next step in the mathematical induction is to allow N two equal to K plus one and when N is equal to K plus one we get the function P of K plus one is equal to K plus one cubed plus two times K plus one. Now we need to go ahead and simplify this statement. In order to simplify the statement we're going to first foil K plus one cubed and we're going to distribute to into K plus one. This is going to give us K cubed plus three K squared plus three K plus one plus two K plus two. Now I'm gonna go ahead and reorder these statements these terms real quick and I'm gonna reorder them in the way that I have K cubed plus two K plus three K squared plus three K. And we're going to combine the like terms which is two in one. So I'm gonna have plus three at the end of the statement. Now the reason why I wrote it like this is because notice that the first two terms K Q plus two K is equivalent to the previous statement in the second step. That means that we can go ahead and substitute this portion of the statement with three R. And that's going to give me three are plus three K squared plus three K plus three. And notice that every term here has a coefficient of three, meaning that we can go ahead and factor out a three from every term in the statement. And this is going to give me the function P of k plus one is equal to three times R plus k squared plus k plus one. Now this statement shows that this entire function is divisible by three. And that's because no matter what we get in this value here, that value is always gonna be multiplied by three, which means that this statement is always going to be divisible by three. And that shows that the original statement three is a factor of n. q plus two. N is true. And that means that the answer to this problem is going to be D. So I hope this video helps you in understanding how to perform mathematical induction and I'll go ahead and see you all in the next video

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n^2 - n.

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