yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Writing arithmetic series in sigma notation


3m read
·Nov 11, 2024

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.

More Articles

View All
The Monkeys of Sri Lanka | Born Wild: The Next Generation
The tropics of South Asia are teeming with wildlife of all kinds. National Geographic Explorer Sandesh Qader travels to a magical setting in Sri Lanka, where different species of baby monkeys learn to navigate life in a complex community. I’m an incredibl…
A Warning For The 2023 Housing Market
What’s up, Graham? It’s guys here, and the housing market is continuing to get worse. For example, you now need to make more than a hundred and seven thousand dollars a year to afford the average home in the United States. Home builder sales have collapse…
Sinking Ship | Wicked Tuna: Outer Banks
Yeah, what time you thinking about pulling the plug? Damn, sir Englishman tonight. You know, whenever you guys won’t leave, I’ll leave. We funny good. I’ll do whatever you want to make. Then we gotta flee in the day. Now look today for the reels of Fort…
Transforming Human Poop Into Eco-Friendly Fertilizer | Best Job Ever
I’m gonna go get in my poop dumping uniform. See you in a minute. Working with poop in Haiti may not necessarily seem like something you could really put your heart into, like a job you could really love. Okay, I’m ready! Yes, but basically we’ve create…
Capturing Death - What One Photographer Learned on Assignment | Exposure
How do you want to die? Is really the question. You know, what is the quality of your death? What is the quality of a good death? It is the thing that we’re most afraid of. You’re going to die. You will be no more. Who, whoever it is that you believe you …
One Man’s Walk in the Snow Creates a Giant Masterpiece | Short Film Showcase
What happened? I didn’t have it in it. Good, did it? I did it! Yes, yes, yes! I suppose inspiration is what comes before motivation. Seeing a good idea gave me that feeling: this could return to something really, really good. There are still things that …