In Exercises 43–54, express each sum using summation notation. Us...

Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

Accepted Answer

Hello. Today we are asked to represent the following some using summation notation and were asked to represent the index and lower limit of the summation as K- & one. So what we are given is 1/4 plus 2/5 plus 3/6 plus all the terms going to 20/20 plus three. Now notice that the final term has a has a sum in its denominator and the sum is exactly the same as the numerator plus three. So if this pattern follows with every other term that means that we're going to have the some or we can represent the sum as 1/ plus three plus 2/2 plus three plus 3/3 plus three plus all of the terms going to 20/20 plus three. So let's just double check this 1/1 plus three is going to give us 1/4 to over two plus three is going to give us two or five and 3/3 plus three is going to give us 3/6. So the pattern follows exactly the same. Now remember what is given to us is that we want to represent the index as K. And one. So since we're representing the index As K&1, we're going to be plugging in all the values of K starting from one and going all the way to the final term. Now if we take a look at the first term, what we are given is 1/1 plus three. If we allow K To equal to one, what we have is one in the numerator over 1-plus 3. And if we take a look at the second term we will have to over two plus three where K is equal to two. So if K is equal to two, we're plugging into into the numerator and plugging in two plus three in the denominator. So if this follows for all the other terms, that means the pattern to this sum is going to be K over K plus three, which means we're plugging in our index into the numerator and the denominator is going to be the index plus three. So that means that the starting point or the lower limit of the index is going to be K is equal to one. And for the final term, 20/20 plus three the final term is where K is equal to 20. So the upper limit or final value of the index is going to be K is equal to 20. Now let's go ahead and plug in everything into the summation notation. We first start by writing our sigma value or sigma and the lower limit is gonna go underneath sigma, which we said is going to be K is equal to one. Since that's the starting value of the index. And on the top of the sigma we're going to put the final value for the index which in this case is going to be 20. And then finally we put our pattern which is going to be K over K plus three, and this is gonna be the summation notation of the given some. 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 find the summation notation, and I'll go ahead and see you all in the next video.

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

Key Concepts

Summation Notation

Index of Summation

Sequence and Series

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