In Exercises 11–24, use mathematical induction to prove that each...

Question

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

1 + 3 + 5 + ... + (2n - 1) = n^2

Accepted Answer

Hello. Today we're going to be proving that the given statement is true using mathematical induction. So what we are given is two plus eight plus 14 plus all the terms to the end term six and minus four. And that is going to be equal to three, N squared minus n. Now the first step in showing whether this is true or not, is to show that three N squared minus N is at least equal to the first term of the summation too. So the first step step in mathematical induction is to let n equal to one and letting N equal to one is going to give us the statement two is equal to three and squared minus N. Now we're going to substitute one for N and that's going to give us two is equal to three times one squared minus 112. Any exponents is just going to be itself one and three times one is just going to give us three. So we have the statement two is equal to three minus one and three minus one is going to give us two which shows that the given statement for n is at least equal to the first term of the summation. Now, mathematical induction states that we want to go ahead and change the statement into a statement that's at least equal to every term in the summation. So the second step in mathematical induction is to allow N two equal to K and letting an equal to K is going to give us the statement two plus eight plus 14 plus all of the terms 2, 6 K now minus four. And that is going to equal to three K squared minus K. So this is the form of the statement that is at least equal to every term in the original summation. So we're going to assume that this statement is true for now and now we want to show that this statement is equal to the summation plus the next term and the next term is going to be one, N is equal to K plus one. So now we want to go ahead and extend the summation by one more term and doing this is going to give us two plus eight plus plus all of the terms 26 k minus four plus the next term, which is going to be six times K plus one minus four. And that is going to equal to three times K plus one squared minus k plus one. So instead of letting n equal to k, we're letting n equal to K plus one. That's why these statements look similar to each other. Bar slightly different. Now, what we want to do with the mathematical induction is to show at this whole left hand side is equal to this right hand side. But how do we do that if we don't even know how many terms are on the left hand side? Well, the tricky thing is to understand this, notice that this portion of the left hand side of the statement is equivalent to the statement we came up with in step two with that being said, we can use this statement as a substitution for this portion in the third step so we can set this entire thing equal to three K squared minus K. And doing this is going to give us three k squared minus K plus while we're going to distribute six into K plus one. So we have the quantity six K plus six minus four and that is going to equal to three times K plus one squared minus K plus one. And before I move on I'm just gonna go ahead and combine the like terms of six and four and six minus four is going to give me too. So I have six K plus two. Now we want to go ahead and show that the left hand side is equal to the right hand side and at this point it's just going to be algebra. So let's go ahead and start by simplifying the left hand side on the left hand side, we're given three K squared minus K Plus six K Plus two. And combining like terms, is going to give us three K squared plus five K plus two. And that's going to be the left hand side simplified. Now we want to go ahead and simplify the right hand side on the right hand side, we are given three times K plus one squared minus the quantity K plus one. And in order to simplify this, we're going to need to foil out K plus one squared and we're going to distribute negative one into the quantity K plus one doing this is going to give me three times K squared plus two K plus one minus k minus one. Now I'm going to distribute three into the quantity K squared plus two K plus one. And that's going to give me three K squared plus six K plus three minus k minus one. And the last step to simplify here is to go ahead and combine the like terms, we can combine six K with negative K and three with negative one. And that's going to give me three K squared plus five K plus two. And that's going to be the right hand side simplified. Now putting this back into the statement, we have three K squared for the left hand side plus five K plus two. And that is going to equal to the right hand side, which is three K squared plus five K plus two. Since both the left hand side and right hand side are exactly the same. This shows that the original statement is equal to every term 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 do mathematical induction and I'll go ahead and see you all in the next video

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n^2

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