Writing arithmetic series in sigma notation

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 eleven, and we keep on adding all the way up to four hundred five.

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 eleven. 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 four hundred five. So let's think about how many times we are going to add two to get to... Sorry, how many times do we have to add two to get to four hundred five?

So four hundred five is seven plus two times what? So let me write this down. If we wanted four hundred five, it 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 four hundred five.

So that is going to be equal to... Let's see, if we subtract seven from both sides, we have three hundred ninety-eight is equal to two X. Or let's see, divide both sides by two, and we get this is going to be what? One hundred ninety-nine. One hundred ninety-nine is equal to X, so we're essentially adding two one hundred ninety-nine times.

So this is the first time we're adding two, this is the second time we're adding two times one, adding two times two, and here we're adding two times one hundred ninety-nine to our original seven.

So let's think about this a little bit. So this is going to be a sum—a sum from... So there are 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 one hundred ninety-nine 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 plus two times K. When K equals zero, this is just going to be seven. When K equals one, it's seven plus two times 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 eleven.

And all the way, when K is equal to one hundred ninety-nine, it's going to be seven plus two times one hundred ninety-nine, which is three hundred ninety-eight, which would be four hundred five. 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 equals one, then let's see, it's going to be... The first term is going to be seven plus two 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, you're going to add two one time because two minus one is one.

So that gives us that one. And so how many total terms are we going to have here? Well, one way to think about it is I just shifted the indices up by one. So we're going to go from K equals one to two hundred.

And you can verify this—when K is equal to two hundred, this is going to be two hundred minus one, which is one hundred ninety-nine. Two times one hundred ninety-nine is three hundred ninety-eight plus seven is indeed four hundred five.

So when K equals two hundred, that is our last term here. So either way, these are legitimate ways of expressing this arithmetic series using Sigma notation.