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. 2+2^2+2^3+⋯+ 2^11

Accepted Answer

Hello. Today we're going to be representing the following some using summation notation. And we're going to represent the index as K. and one. So since we're representing the index as K&1, that means that our starting value for K is going to equal to one. And we're going to be plugging in all the values following one all the way until the final value for K. Now let's go ahead and figure out what the final value of K is and what the pattern for the summation is by taking a look at the given some. So the first term of the sum is just six. The second term is six to the power of two and the third term is six to the power of three. Notice that there's a slight pattern here, the base of all the exponential values of all the terms are going to be the same. It's just six. But notice that all the exponents are increasing by one for the first term. Six just has the exponents of one. So since the exponents follow the pattern of 123 and our index represents the pattern of 12 and three. That means that we're plugging in the value of the index for the exponent of six. And that means that the pattern to our summation notation is going to be six to the power of K. Now let's go ahead and figure out what the final term for the index is going to be. The final term here is six to the power of 20. And following this pattern, K would have to equal to 20 in order for us to get six to the power of 20. So that means that the final value of the index is going to be Since we have the lower limit which is K is equal to one. And the upper limit which is K is equal to 20. Let's go ahead and put this into the summation notation. We start that by drawing our sigma symbol which looks like this. The lower limit goes at the bottom of the sigma symbol, which is gonna be the starting index value. And that's going to be K is equal to one. And the final value for K goes at the top of Sigma, which in this case is going to be 20. And finally we put the pattern to the right of sigma and that's going to be six to the power of K. 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 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. 2+2^2+2^3+⋯+ 2^11

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