In Exercises 61–68, use the graphs of and to find each indicat...

Question

In Exercises 61–68, use the graphs of and to find each indicated sum.

Two side-by-side scatter plots showing sequences an and bn with points plotted for n values 1 to 5 on a grid.

$\sum_{i = 1}^{5} a_{i}^{2} - \sum_{i = 3}^{5} b_{i}^{2}$

Accepted Answer

Hello. Today we're going to be using the given graphs to evaluate the given summation. Now, before we take a look at the summation, let's go ahead and take a look at the given graphs. The graph on the left depicts every value for the sequence A sub N and the graph on the right depicts every value in the sequence B sub N. Now, before we jump into the summation, I want to go ahead and label each of the given values. And the way that we read this is that we want to take a look at the corresponding and value which the first point here is going to be N is equal to one and see what value it gives us in return. Well, when N is equal to one for the A graph, we're going to get the returned value of four. And what this means is that the first term in the sequence A which is a sub one Is going to equal to four. And we want to go ahead and repeat this pattern for the remaining points. So when N is equal to two, we get the return value of one. So we're going to have a sub two is equal to one. When N is equal to three, we get the return value of negative two. So a sub three is equal to negative two. And finally, when N is equal to four, we get the return value of negative five. So a sub four is equal to negative five. Now we want to go ahead and repeat this process for the B graph. So for the B graph, when N is equal to one, we get the value of negative five. So B sub one is equal to negative five. When N is equal to two, we get negative two. So B sub two is equal to -2. When N is equal to three, we get one. So B sub three is equal to positive one. And finally, when N is equal to four, we get the return value of four. So B sub four is equal to four. Now that we have all the terms listed, let's go ahead and take a look at the given summation. Now, the given summation states that we want to take the sum of the first four terms of the A sequence. But every term is going to be squared, then we're going to add that to the summation of the first four terms of two times every term in the B sequence cubed. Now, since we're dealing with to the sum of two summations, let's go ahead and evaluate each summation individually. So I'm going to label the a summation as one and I'm going to label the B summation as two. And so for the first one, what it's stating is that we're going to take every term in a square it and add it together. So the first term is when N is going to equal to one and that's going to give us a sub one which is four, but we have to square that term. So we get four squared, then we're going to add the next term when N is equal to two. And so a sub two is equal to one squared. Next we add the next term which is when N is equal to three and that's going to give us a sub three squared which is going to give us negative two squared. Finally, we're going to add the term when N is equal to four. So that's going to be a sub four squared, which is going to give us negative five squared. Now let's go ahead and do some simplification. Four squared is going to give us 16, 1 squared is just going to give us one negative two squared is going to give us positive four and negative five squared is going to give us positive 25. So 16 plus one plus four plus 25 is going to give us a total sum of 46. And that's what the first summation is going to equal to. Now let's go ahead and evaluate the second summation. The second summation states that were taking every term in be multiplying it by two and then cubing it. So what that's going to give us is when N is equal to one, we have B sub one is equal to negative five. So that's going to give us two times negative five that quantity cubed. Then we're going to add one N is equal to two. That's going to give us B sub two, which is equal to negative two. So we have two times native two cubed. Next, we're going to add one end is equal to three. And that's going to give us B sub three which is equal to positive one. So we have two times one quantity cubed. Then finally, we want to add the term when N is equal to four and that's going to give us B sub four which is equal to four. So that's going to give us two times four quantity cubed. Now let's go ahead and do a little bit of simplification. Two times negative five is going to give us negative 10. So we have negative 10 cubed plus two times negative two, which is going to give us negative four cubed Plus two times 1, which is going to give us two cubed Plus four times 2, which is going to give us a cubed negative 10 cubed is going to give us negative 1000. Then we're going to add negative four cubed, which is going to give us negative 64 plus two cubed, which is going to give us eight plus eight cubed, which is going to give us 512. So if we add negative 1000 with negative plus positive eight plus 512 that's going to give us a total sum of negative 544 which is what the second summation is going to equal to. So now that we've evaluated both of the summations individually, let's go ahead and plug in the values back into the original statement. And what this is saying is that's the first summation, which is 46 plus the second summation, which came out to be negative 544. And finally, 46 plus -544 is going to give us a total sum of -498. And that's going to be the evaluated form of the given summation. And with that being said, the answer to this problem is going to be be. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 61–68, use the graphs of and to find each indicated sum.

$\sum_{i = 1}^{5} a_{i}^{2} - \sum_{i = 3}^{5} b_{i}^{2}$

Key Concepts

Sequences and Terms

Summation Notation (Sigma Notation)

Evaluating Sums of Squares of Sequence Terms

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