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.

n + 2 > n

Accepted Answer

Hello. Today we're going to show that the following statement is true using mathematical induction. So the first step in mathematical induction is to show that the given statement is true when n is equal to one and when n is equal to one, we get the statement one plus four is greater than plus four is going to give us five. And it is true that five is greater than one. So the first step of the mathematical induction is true. Now the second step of the mathematical induction is to allow end to equal to K. And when N is equal to K, we get the statement K plus four is greater than K. Now the purpose of this statement is to show that any integer K is always going to make this statement true. So we're going to assume that this statement is true for now. And finally the third step is to show that the statement is true when n is equal to K plus one and when n is equal to K plus one we get K plus one plus four is greater than K plus one. So now we just need to simplify this statement. One plus four is going to give us five. So we have K plus five is greater than K plus one. And finally we can just subtract one to both sides of the inequality. And that's going to give us k plus four is greater than K. And notice that this statement is the same thing as the second statement. And we said that the second statement is true for any integer K. So this shows that the original statement, and plus four is greater than N, is true for any integer N. And with that being said, 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. n + 2 > n

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