In Exercises 10–11, express each sum using summation notation. Us...

Question

In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17

Accepted Answer

Hello. Today we're going to find a proper summation notation for the given series. So the series given to us is 4/7 plus 5/8 plus 6/9 plus dot dot dot plus 17/20. So looking at the summation, we're gonna have some type of summation where the starting point we can assume to be one, but we don't know the endpoint and we know that the argument needs to be in some form of a fraction because all of the terms here are fractions. So let's go ahead and try to figure out how to represent the numerator. So if you look at the summation here, Each numerator is increasing by an increment of one. So our first term is four And if we do 4 -1, that's just going to give us three. So we can assume that the numerator is going to be I plus three. So for now let's just go ahead and put that in our assumption and we're gonna do the same thing for the denominator, the denominator increases by an increment of one. So if the denominator in our first term is 77 minus one is gonna give us six. So the denominator needs to be represented by I plus six if the starting point is i is equal to one, so we have I plus six in the denominator. Now we need to figure out what our end term is going to be. And in order to do that we can go ahead and take our summations which is I plus three and I plus six. And set it equal to the numerator and denominator of the final term. So the assumption for the numerator is I plus three and the numerator of the final term is 17. So setting this equal to 17 we can go ahead and sulfur I which will be the final term and 17 minus three is going to give us 14. Now let's go ahead and double check this with our denominator. So we know that the denominator is equal to I plus six and we're gonna go ahead and set that equal to 20. So I plus six is going to equal to 20 and 20-6 is going to give us 14, which tells us that the confirmed end point of the summation is going to be 14. So our summation that properly represents this series is gonna be the summation from when is equal to 1 to 14 of I plus three over pi plus six, which means that the solution to this problem is going to be C. So, I hope this video helps you in understanding how to find a serious representation for a summation. And I'll go ahead and see you all in the next video

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