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### Course: Algebra (all content) > Unit 18

Lesson 6: Advanced sigma notation- Arithmetic series in sigma notation
- Arithmetic series in sigma notation
- Finite geometric series in sigma notation
- Finite geometric series in sigma notation
- Evaluating series using the formula for the sum of n squares
- Partial sums intro
- Partial sums: formula for nth term from partial sum
- Partial sums: term value from partial sum
- Partial sums intro

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# Arithmetic series in sigma notation

Sal writes the arithmetic sum 7+9+11+...+403+405 in sigma notation. There are actually two common ways of doing this.

## Want to join the conversation?

- How about (k=1) on bottom, 200 on top, and (5+2k) to the right of the sigma?(9 votes)
- That is another way of getting the same arithmetic series. Clever thinking!(4 votes)

- express the series in sigma notation

2+3+4+9+8+27+16+81(2 votes)- I am not an expert and I could have the wrong answer anyway.

I am going to give you a hint: notice the numbers they seem familiar, did you figure it out ?

they are the powers of 2,3 added together !

so they can be written like this:

4

Σ ( 2^x + 3^x )

x=1(1 vote)

- At3:25, what did Sal say 2 - 1 is 2...? I got confused by that. Was it an error or it actually has something to do with the calculation? Or he probably meant that when K=2 in 2(k-1)....2 minus 1 is 1, times 2 which equals to 2...?(2 votes)
- Two questions: What does n mean? Doesn't it symbolize the term number? And:

If I have the infinite arithmetic series 8+12+16+20+..., how would my sigma notation look like? For this example, would 8 be n=1, 12 be n=2, 16 be n=3,...?

I have that n=1 and goes to infinity with the expression 8+4(n-1).

But my textbook has a different answer. It says n=2 and goes into infinity with the expression 4n. But why n=2? That means that my second term (it being 12) somehow equals to 8.

I really hope someone will understand what I mean..Pls help!(1 vote) - How would you get the answer to a problem where you already have the sum with the terms after it...

8

Σ -2+5n

n=2

How would you solve that? I am just completely confused...what am I solving for?(1 vote)- You're trying to find the value of the equation. It wants you to find out what the terms from the second term to the eighth term add up to.

1.Plug in the numbers from 2 to 8 for n (so plug in 2, 3, 4....all the way to 8.

2. add the numbers together (so if you plugged 2 in, you would get 8, and then if you plugged in 3, you would get 13, and so on until 8, and then you would add those numbers together)

3. now you've found the value! (it's 161)

This got me confused, too. Thank you, actually, for asking this; it made me learn the lesson better because I wanted to help you out. Hope this helps!(1 vote)

- I am still easily mixed when it's the right situation to add the 1st value. For instance, in this video, when k=1, in the 2nd expression, the 1st value will still be "7", which is already counted in the sum, why the n becomes 200 instead 199?(1 vote)
- or how about sum(2k+1), k=3 to 202(1 vote)

## Video transcript

- [Voiceover] What I
want to do in this video is get some practice writing
series in sigma notation and I have a series in
front of us right over here. We have seven plus nine plus 11 and we keep on adding
all the way up to 405. So first, let's just think
about what's going on here. How can we think about what
happens at each successive term? So we're at seven and
then we're going to nine and then we're going to 11. It looks like we're adding two every time so it looks like this
is an arithmetic series. So we add two and then we add two again and we're going to keep adding two all the way until we get to 405. So let's think about how many times we are going to add two to get to 200, sorry, how many times we have
to add two to get to 405. So 405 is seven plus two times what? So let me write this down. So if we wanted 405 is equal to seven plus two times, I'll just write two times x. I'm just trying to
figure out how many times do I have to add two
to seven to get to 405? And so that is going to
be equal to, let's see, so we subtract seven from both sides. We have 398 is equal to two x or let's see, divide both sides by two and we get this is going to be what? 199? 199 is equal to x. So we're essentially adding two 199 times. So this is the first
time we're adding two. This is two times. We're adding two times
one, adding two times two and here, we're adding two times 199 to our original seven. So let's think about this a little bit. So this is going be a sum, a sum from, so there's a couple of ways
we could think about it. We could think about how
many times we've added two so we could start with
us adding two zero times, the number seven is when we
haven't added two at all, all the way to when we add two 199 times. And let's think about this a little bit. This is going to be, we could write it as
seven plus two times k. Seven times two, seven plus two times k. When k equals zero, this
is just going to be seven. When k equals one, it's
seven times two plus one. Well, it's going to be nine. When k is equal to two, it's going to be seven plus
two times two which is 11. And all the way when k is equal to 199, it's going to be seven plus
two times 199 which is 398 which would be 405. So that's one way that we could write it. Another way, we could also write it as, let me do this in a different color, we could, if we want to start our index at k is equal to one then let's see, it's
going to be the first term is going to be seven plus two times k minus one, times k minus one. Notice, the first term works out because we're not adding two at all so one minus one is equal to zero so you're just going to get seven. Then when k is equal to
two, the second term, we're going to add two one time because two minus one is two so that gives us that one. And so how many total terms
are we going to have here? Well, one way to think about is I just shifted the indices up by one so we're going to go
from k equals one to 200. And you can verify this. When k is equal to 200, this is going to be 200
minus one which is 199. Two times 199 is 398 plus seven is indeed 405. So when k equals 200, that
is our last term here. So either way, these are legitimate ways of expressing this arithmetic series in using sigma notation.