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Hello everyone. So in this video, I'd like
to talk about summations and products,

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but first I'd like to review 
what the notation means. So

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recall that when we have a 
sum from i equals 1 to n of a i,

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this is literally the terms

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when you...you start by 
plugging in i equals 1,

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you get the term a1,

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then you increment i
so you go to i equals 2.

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If 2 is less than or equal to n, you 
plug it into here and you repeat

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until you get to the very end, so 
a1 plus a2 all the way up to a n.

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Again if n, for example, was 1,

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there will only be one term in the sum.

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If we change this sum to the product, the only 
thing that changes is the operation between them,

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so instead of sums, they're products now.

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With this in mind, let's answer 
the following questions,

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so question 1: "Write the following summation
using a summation starting from 1."

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So here the summation 
starts at 3 goes up to 28,

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and you have this expression here.

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So how do we change 
the bottom index

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to start from 0? Well the idea

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is

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you're going to use

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a new letter. So we're going to 
define a new letter, let's call it j,

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and we're going to define it to be i minus 2.

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Now you can rewrite the summation using the same 
letter but I'm going to use a different letter just

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for ease of notation.

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So now that j is i minus 
2, when i is 3, j is 1

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and when i is 28 j is [26] so...

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if I replace i 

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with j plus 2 

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and write up the summations, 
let's see what we get.

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Let's see what we get. So when 
we plug it into the summation,

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i equals 3 becomes j equals 1,

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28 becomes 26 that's [28] minus 2,

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the i becomes j plus 2, and the i 
plus 1 becomes j plus 2 plus 1,

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so let's write that as j plus 3. Let's make it

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in its simplest form.

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And these two summations are equal. Right?

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This is what you call substitution. You could 
substitute a letter for a different letter,

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and you can modify your summation

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accordingly.

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Now, for example,

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you can also say that these two 
things are equal. We can also just

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expand some of the terms out 
which might be a good exercise so

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I'm just... let's call it a "for fun" section.

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So the sum

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is going to be 3 times 3 plus 1

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to the power of 3.

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So if we start to evaluate this sum, right, the 
first term is 3 times 3 plus 1 cubed, right,

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I'm just plugging in 3. I 
increment 3 and I go to 4,

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and I get the next term:

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the 4 times 4 plus 1 cubed.

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Now you increment i to 
go to 5, 5 is less than 28,

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and I continue and I
continue and I continue.

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When's the last term? Well the last term is going 
to occur when i equals 28 so I'm going to get 28,

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28 plus 1 cubed.

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So hopefully we all agree that that's true

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and then we can do this 
with the other sum as well.

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Again, this is just for practice; just 
to make sure we understand

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the notation and that we can actually do this.

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So if I see the other sum,

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I'm going to plug in the first 
term which is 1 plus 2,

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1 plus 3 cubed,

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plus 2 plus 2, 2 plus 3

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cubed,

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and then the last term, again 3, 2, 3… 

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so I plug in 1, I plug in 2, I plug in 
3, and I go all the way up to 26.

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So let's see what we have, 
sorry that was a typo,

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so we plug in the 1 term into 
here so 1 plus 2, 1 plus 3.

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Hopefully we agree that 
those two things are equal,

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then we have 2 plus 2, and 2 plus 3

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so 4 and 5, those two things are 
equal, and then we keep going

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and at every step, even though 
I didn't show all the steps,

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28 is 26 plus 2, 28 plus 1 
is the same as 26 plus 3.

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These two sums are actually 
equal, every term is the same.

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Okay?

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Hopefully this gives you a idea of 
how to change the indices so that

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you can start it at whatever number you'd like,

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and this might be important 
for certain applications.

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Question:

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"Write the sum from i equals 
k plus 2 to 3k plus 1 i

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as a summation from i 
equals k to i equals 3k

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of i." 

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You want to use other terms as well, right, because 
there's no way these two things are going to be equal.

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So let's see that now.

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Let's look at this, so

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I'm not going to expand this sum 
out like I did the last one, just for

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sanity's sake.

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We should be able to do this just by looking 
at it, right? Where do these differ? Well…

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let's look at this equality.

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Now the terms here, there's a couple 
of extra terms. Right? We start at k,

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but this sum started at k plus 2, right? 
So this one's going to have a k,

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a k plus 1, a k plus 2, k plus 3 
all the way up to 3k, right,

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but the k and the k plus 1 
terms aren't the same.

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So if I'm going to include the
k and k plus 1 terms in the sum,

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I'd better remove them.

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Right now, if you expand this out, 
this is a sum from k, k plus 1,

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k plus 2, k plus 3, but now I’m
subtracting off k and k plus 1.

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So I could condense this to 
a sum starting at k plus 2,

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so these two things at the
bottom end are equal. 

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Now on the top end though, this one has 
a term 3k plus 1 and this one stops at 3k.

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So to account for that, I need to 
include the 3k plus 1 term. If I plug in

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3k plus 1 in for i, I'm just 
going to get 3k plus 1.

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You could imagine changing this question slightly.

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For example, if I made
everything squared,

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instead of i I had i squared, 
then I would instead of just

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k and k plus 1,

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I would have squares everywhere.

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So let me do the changed question.

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Sorry, I didn't actually put up the answer 
to the other one, sorry. It didn't render.

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So what if I changed it to, "Write i
squared as a summation",

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so write this sum as a summation 
like this with other terms.

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Here again, I'm starting from 
i equals k to 3k of i squared.

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The kth and the k plus first terms,

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these are

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extra on this side,

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right, this starts at k plus 2 but 
this starts at k, so I need to

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remove those in order to get the equality on 
the bottom end of these two sums to be equal.

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And on the top end, well this one

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finishes at 3k plus one and
this one finishes at 3k

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so I need to add in that 
extra term that I've lost.

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Now, for example, if I change 
the 3k plus 1 to a 3k plus 2,

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then I would just simply
add the extra 3k plus 2 term.

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And there you have it. 

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These are all ways to manipulate summations,

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these aren't all of the ways but 
these are some of the ways to do so.

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So hopefully this gives you a better 
understanding of the notation

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and helps you out for your
midterm. Best of luck.