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JH
Numerade Educator

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Problem 16 Easy Difficulty

(a) Explain the difference between
$ \displaystyle \sum_{i = 1}^{n} a_i $ and $ \displaystyle \sum_{j = 1}^{n} a_j $
(b) Explain the difference between
$ \displaystyle \sum_{i = 1}^{n} a_i $ and $ \displaystyle \sum_{i = 1}^{n} a_j $

Answer

(A). No difference
(B). $\sum_{i=1}^{n} a_{i}=a_{1}+a_{2}+\dots+a_{n}$ whereas $\sum_{i=1}^{n} a_{j}=a_{j}+a_{j}+\cdots+a_{j}=n \cdot a_{j}$

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Video Transcript

Let's look at part A and let's explain the difference between these two summations. So let's look at the first one. This is the sum I equals one and a I. And let's actually just write that out. This means that you add up the A's and you plug in one for I. First you plug into and you keep going all the way until you get to the final. And however here you can see by this expression on the right hand side. This does not depend on II. No, does not depend on I this right hand side here. So really, if we go ahead and replace this eye with a J, that's not going to change the answer because it's when we write it out. We're still going to get this sum from one to the end. So is forget. If we use the J down here, we must make sure that we're calling this J under the as well. Here. We need peace to match because this index, the under the A. That's the thing that's changing each time, and so therefore, if you want the these numbers to change under the A's as you some you want the thing under the A to be the same as the thing down here under the some same letter J. So these are they're people. There is no difference from these two sums and party, eh? However, if we look at part B So let's separate this from our party. The sums don't quite. It's not just that they went for my deejay, but here they kept the bottom both in terms of I. So the left hand some. We already wrote this. Let's just write it again. So you plug in. I so notice it's the same. I hear I there. So we would just add them all up one all the way up, Tio and everything in between. However, for the second one, this is exactly what we were talking about in part A. When does not what should not happen? So here we are, adding a J. But this some does not depend on J. So here, if you go ahead and plug in, I equals one. Well, there's no I replied in this expression does not depend on I. So this expression here you could say the same thing. So when I plug in I equals one. Well, I'm just going to get out a J because there's nothing to do this doesn't depend on. And similarly, if you plug in, I equals two. You get a J all the way up to when you plug an end. So this is we have a J and is showing up and times one for each of the eye values we plug in one all the way to end. So here's I equals one eye goes to They're all the way that I equals And these were the terms that we that of course fund. So we just have end times a J. So that's the difference between these two sums. What? You actually have a sum of all the numbers a one through. And on the other hand, this sum over here is just end time some number A J that they decided to put here in front of the sun. And that's your final answer