Use mathematical induction to prove that the statement is true fo...

Question

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

Accepted Answer

Welcome back. I'm so glad you're here. We've got a statement that we get to prove. Using mathematical induction. So this is fun. We first we recall from previous lessons that the first step of mathematical induction is to show that S sub one is true. Which means that we're going to take anywhere. There's an N. And substitute a one for our number. So using this part here from the left will have instead of seven and we'll have seven times one equals and then on the right hand side instead of seven. And we've got seven times one times instead of N plus one. It's one plus one all divided by two. And so now we can simplify that on the left hand side. Seven times seven. Well that are seven times one. That's just seven. And on the right hand side and the numerator, we've got seven times one. That's seven times one plus one is two. All divided by two. Will the twos cancel out? And the right hand side is also seven. So yes, we have shown that as someone is true. Great. We can move on to step two. Step two has two parts. The first part is to assume that S sub K. Is true and that means we're going to go and stick a K in anywhere that there was an end. So we're going to copy the whole thing this time we go seven plus 14 plus 21 plus 28 plus all the rest plus seven. N. Excuse me. So N. Is where we start substituting the K. So plus seven K equals. Now instead of seven N. It's seven K. Times instead of N plus one. It's K plus one. All divided by two on the right hand side. Okay that's the thing we're assuming is true and because we've assumed that's true now we're going to prove that S sub K plus one is true. That's the second half of the second step. So now S sub K plus one meaning in the original where there's an end we're now going to substitute K plus one so we'll copy this down. Seven plus 14 plus 21 plus 28 plus everything in between plus seven K. We're keeping that because that's the number preceding seven K. Plus one. Our new one and that equals seven instead of K. It's now K plus one times instead of K. There it's K. Plus one plus one. And that's all over two. And now here's the fun part. With mathematical induction we know that this chunk down here that we copied is the same as this chunk preceding it. Seven plus 14 plus etcetera two plus seven K. And that left hand side and the S. F. K. Is the same as the right hand side that equal. So we can substitute this part the seven K. Times K plus one divided by two in for this chunk. And the left down here of our S sub K plus one. So what does that look like? That means? Now we've got seven K. Times K plus 1/2. And now we just bring down this part plus seven times K plus one. And now that all equals seven times K plus one times K plus one plus one, that's two all over to. And now this is just algebra so we can take the whole thing and multiply it all by two. All the numerator is get multiplied by two. And so in that first term that is all going to cancel out the two in the denominator. So it's just seven K times K plus one. And this um second term here the two gets multiplied by the seven. So it's plus 14 K plus one equals. And over here the two in the numerator and the denominator cancel out. So it's seven times K plus one times K plus two. Great. And now let's just do some distribution. Seven times K. Or seven K times K is seven K squared, seven K times one is plus seven K plus 14 times K. Is 14 K plus 14 times one is 14. On the right hand side, seven times K. Seven K plus seven times one is seven. And then we'll bring down that K plus two. We will foil that in a minute. So on the left hand side we've got seven K squared plus 21 K combined like terms plus 14 equals. And now we will foil this seven K times K. Seven K squared seven K times two is plus K. seven times K is plus seven K, And seven times 2 is plus 14. Simplifying one more time. On the right hand side is seven K squared plus 21 K plus 14. And look at the left hand side, seven K squared plus 21 K plus 14. They match. We have proved that S sub K plus one is true and therefore we can choose answer choice D. The statement is true. Well done. We'll catch you on the next one.

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