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.

5Σi=4 (a_i/b_i)^2

Accepted Answer

Hello. Today we're gonna be using the two 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 two graphs given to us, the graph on the left depicts every value in the sequence. A sub N and the graph on the right depicts every value in the sequence B sub N. So what we want to go ahead and do is label the value for every point given in both sequences. So for example, in the case of end sequence, we're going to take a look at every value of N and see what the corresponding ASAP and value is going to be. So for example, when N is equal to one, this is going to give us a sub end value of positive four. And what that means is that a sub one is going to equal to positive four, then we're going to repeat this process for every other value of N. So when N is equal to two, we get the value of one. So a sub two is equal to one. When N is equal to three, we get the value of negative two. So a sub three is equal to negative two. And finally, when N is equal to four, we get the value of negative five. So a sub four is equal to negative five, then we're going to repeat this process for the B sub N graph. So when N is equal to one, we get the value of negative five. So B sub N is equal to negative five. I'm sorry. B sub one Is equal to -5. Then when N is equal to two, we get the value of negative two. So B sub two is equal to negative two. When N is equal to three, we get the value of one. So B sub three is equal to positive one. And finally, when N is equal to four, we get the value of positive four. So B sub four is equal to positive four. Now that we have all the values labeled, let's go ahead and take a look at the given summation. So the summation here states that we want to add the first four terms of the, every term in B divided by every term in A, that value squared. So what that means is that the first term in the sequence is going to be B sub one divided by a sub one that quantity squared B sub one is negative five and A sub one is four, then we're going to square that value and add the next term the next term is going to be B sub two over a sub two. Quantity squared B sub two is negative two and A sub two is one. And that quantity is going to be squared. Next, we're going to add B sub three divided by a sub three. Quantity squared B sub three is positive one and a sub three is negative two. And that quantity is squared. Finally, we're going to add B sub four divided by a sub four, that quantity squared B sub four is positive for and a sub four is negative five. And that quantity is going to be squared. So now we have a bit of simplification to do using the power distribution rule, we can distribute the exponent of two to each of the numerator and denominators in our given sequence. And doing this is going to give us negative 5/4 quantity squared, which is going to be 25/16 plus negative 2/1 quantity squared, which is going to give us 4/1 plus one over negative two quantity squared, which is going to give us 1/4 plus four over negative five quantity squared, which is going to give us 16/25. Now that we've simplified the exponents, we want to go ahead and add the fractions together. But in order to do that, we need to have the same denominator in each of the terms. So we're going to need to look for the lowest common denominator between 1614 and 25. And in this case here, the lowest common multiple between these four numbers is going to be 400. So we're going to need to multiply each fraction by a value that's going to give us 400. In the denominator, We can go ahead and multiply the first fraction by 25/25, We can multiply the second fraction by 400 over 400. The third fraction can be multiplied by 100 over 100 And the last fraction can be multiplied by 16/16. So if we multiply the first fraction by 25/25, what we are left with is over 400. If we multiply the second fraction by 400 over 400, what we have is 1600 over 400, multiplying the third fraction by 100 over 100 is going to give us over 400. And finally multiplying the last fraction by 16/16 is going to give us 256 over 400. Now that each of the fractions has the same denominator, we can go ahead and combine this into a single fraction by adding the numerator is together. So what we are left with is in the numerator 625 plus 1600 plus plus 256, which is going to give us a total sum of 2,581 all over 400. And that is going to be the evaluated form of our 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.

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In Exercises 61–68, use the graphs of and to find each indicated sum.
5Σi=4 (a_i/b_i)^2