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+2^2+3^2+⋯+ 15^2

Accepted Answer

Hello. Today we are asked to represent the following some using some nation notation. And we are asked to represent the values of the index as K. and one. So since we are asked to represent the index As K. N. one, that means that the starting value for the index is going to be one. And we're gonna be plugging in all the values afterward all the way into the final value of the index. Let's go ahead and figure out what the pattern and final value of the index is going to be by taking a look at the given terms. So the first term in the sum is one to the power of six. The second term is 2 to the power of six And the third term is 3 to the power of six. The pattern between all the terms here is that every term has an exponent of six. The six is not changing. However the basis of the exponents are changing as the base in the first term is one. In the second term is two and the third term is three. But notice how the 1st, 2nd and 3rd term have the pattern of 123. And the beginning values of the index have 12 and three. That means that we're plugging in the vice of the index for the base of every exponential value for the terms. And with that being said, the pattern to the sum is can be represented as K to the power of six. That means that any value given in the index. We plug in for the base and we raised to the power of six. Now, in order to find the final term or the final value for the index, we're going to take a look at the final term of the sum Which is given to us as 20 to the power of six. In order to get 20 to the power of six and stay respective to the pattern. This is where K is going to equal to 20 because if we take K is equal to 20 and plug it into our pattern, we get 20 to the power of six. So the final value for the index is going to be 20. So that means that the lower limit of the index is going to be K is equal to one and the upper limit of the index is going to be K is equal to 20. Now that we have this information. Let's go ahead and put this into the summation notation. We first start by drawing our sigma symbol. The lower limit is going to go to the bottom of the sigma symbol which is the starting value of the index. And in this case here it's going to be K is equal to one and the final value or upper limit of the index goes at the top of the sigma, which the final value here is going to be 20. Finally we put the pattern to the right of sigma and the pattern is going to be K to the power of six, and this is going to be the summation notation for the given some. And with that being said, the answer to this problem is going to be a. 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+2^2+3^2+⋯+ 15^2

Key Concepts

Summation Notation

Index of Summation

Series of Squares

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