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

Question

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

Graph of sequence a_n with red points plotted on a grid.

Graph of sequence b_n with red points plotted on a grid.

5Σi=1 a_i^2+5Σi=1 b_i^2

Accepted Answer

Hello, today we're going to be using the two given graphs to evaluate the given summation. So before we jump into the summation, let's go ahead and take a look at the two graphs. Given to us, the graph on the left shows every term in the given sequence. A sub N and the graph on the right shows every term in the given sequence B sub N. So let's go ahead and label every term or every point on the given graphs. And the way that we do this is we're going to take a look at every value for N which in this case, for the first point is going to be one and pay attention to what the return value is going to be. So on the A graph, when N is equal to one, the return value is going to be four. And that means that the first term in the sequence A sub N which is going to be a sub one is going equal to positive four. And we're going to repeat this for the remaining points. So when N is equal to, to A sub N is equal to one, so a sub two is equal to positive one. When N is equal to three, A sub three is going to equal to negative two. And when N is equal to four, A sub four is going to equal to negative five. And we're going to repeat this process for the B graph. So on the B graph, when N is equal to one, we get the return 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 negative two. 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, B sub four is equal to positive four. So now that we have all of our points labeled, let's go ahead and take a look at the given summation. So the given summation states that were taking a difference of two summations, the first summation, we're going to add the first four terms of the sequence A and we're taking every term multiplying it by two and then squaring it. And on the right hand side, we're taking the sum of the first four terms of the B sequence, but we're taking every term and cubing it. So since we're dealing with two separate summations, let's go ahead and evaluate the first two summations separately. So I'm gonna label the A summation as one and the B summation as two. So let's go ahead and first evaluate the estimation. So again, the estimation is stating that we're adding up every term multiplied by two squared. So that means that we're going to have two times a sub one which is four squared, then we're going to add two times a sub two which is positive one squared. Next, we're going to add two times a sub three which is negative two squared. And then finally, we're going to add two times a sub four which is negative five, that quantity squared. Now let's do a little bit of simplification. Four times two is going to give us eight. So the first term is going to be eight squared, two times one is going to give us positive two. So we have two squared, negative two times two is going to give us negative four and we have negative four squared and finally negative five times two is going to give us negative 10 and we have negative 10 squared, eight squared is going to give us 64 2 squared is going to give us four, negative four squared is going to give us positive 16 and negative squared is going to give us positive 100. So what we are left with is 64 plus four plus 16 plus 100. And that's going to give us a sum of 184. So that's what the first summation is going to evaluate to. Now let's go ahead and evaluate the second summation. Now again, the second summation is stating that we're adding every term cubed. So that means that we're going to add B sub one, which is going to be negative five cubed plus B sub two, which is negative two cubed plus B sub three, which is one cubed plus B sub four, which is four cubed. And again, for simplification, negative five cubed is going to give us negative negative two cubed is going to give us negative 81 cubed is going to give us positive one and four cubed is going to give us positive 64. So negative 1 25 minus eight plus one plus 64 is going to give us a final total of negative 68. And that's going to be the second summation evaluated. So now that we've evaluated the two summations, let's go ahead and plug it back into the original values. So we said that 184 is going to be what the first summation evaluates to. So we're going to have 184 minus the value of the second summation which came out to -68. So we have negative, we have positive 184 minus negative which can be rewritten as 184 plus 68 And 184 plus 68 is going to give us a final total of 252, which is going to be the summation evaluated. 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 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.

5Σi=1 a_i^2+5Σi=1 b_i^2

Key Concepts

Sequences

Summation Notation

Graph Interpretation

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