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+3+5+⋯+ (2n−1)

Accepted Answer

Hello. Today we are asked to represent the following some in some nation notation and we are told to represent the index using K and one. So what we are given is one plus four plus seven plus all the terms going to the end term of three n minus two. So the final term in the some is called the 10th term. And the term is a representation of every term in the sum. So in order to check to make sure that this is an accurate representation, we can go ahead and take our end term which is three and minus two and set it equal to the first term in the sum, which is one. And since we're comparing it to the first term of the some, we're going to allow end to equal to one substituting end with one is going to give us three times one minus two is equal to 13 times one is just going to give us three and three minus two is going to be one. So we get the statement one is equal to one which is true. So we know that the term at least accurately represents the first term of the sum. But now we want to find the pattern of the summation. And in order to do that we need to find a term that accurately represents every term in the some while in order to do that we can take our end term and allow end to equal to K. And once we allow end to equal to K we get the sum one plus four plus seven plus all of the terms going to the cave term, which is three k minus two and three k minus two is a term that accurately represents every term in the some. So we can say that the pattern for summation notation is going to be 3K -2. Now we want to figure out the upper and lower limits for the index. Now the problem states that they want is to represent the index using K&1. So for the index we're going to allow the lower limit of the index to be K is equal to one because that's the first value of the index. We're gonna be plugging in and this is going to be true for all the values for the for the index going to the end of term. So we're gonna plead plugging in 123 and all the terms to the 10th term, which is going to be represented as N. So again, the lower limit of the index is K is equal to one. And since the last term is the term, the upper limit for the index is going to be K is equal to end. Since we have our upper and lower limit and our pattern, we can go ahead and represent this in summation notation and we start this by first writing the symbols sigma. Now the lower limit goes at the bottom of sigma and we set the lower limit is K is equal to one. So we're going to write K is equal to one underneath the sigma symbol. Now we want to go ahead and write the final term of the index on the upper part of the sigma and that's going to be when K is equal to end. So we're just going to write end And finally we put the pattern to the right of the Sigma symbol, which he said was 3K -2. And this is going to be the summation notation for the given some. And that means that the answer to this problem is going to be D. So I hope this video helps you in understanding how to represent a sum in 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+3+5+⋯+ (2n−1)

Key Concepts

Summation Notation

Arithmetic Sequence

General Term of a Sequence

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