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

Partial sums: formula for nth term from partial sum | Series | AP Calculus BC | Khan Academy


2m read
·Nov 11, 2024

Partial sum of the series we're going from one to infinity summing it up of a sub n is given by, and they tell us the formula for the sum of the first n terms. They say write a rule for what the actual nth term is going to be.

Now to help us with this, let me just create a little visualization here. So if I have a sub 1 plus a sub 2 plus a sub 3, and I keep adding all the way to a sub n minus one plus a sub n, this whole thing, this whole thing that I just wrote out, that is sub sub n. This whole thing is s sub n, which is equal to n + 1 over n + 10.

Now, if I want to figure out a sub n, which is the goal of this exercise, well, I could subtract out the sum of the first n minus one terms. So I could subtract out this, so that is s sub n minus one. And what would that be equal to? Well, wherever we see an n, we'd replace with an n minus one, so it would be n plus 1 over n - 1 + 10, which is equal to n over n + 9.

So if you subtract the red stuff from the blue stuff, all you're going to be left with is the thing that we want to solve for. You're going to be left with a sub n. So we could write down a sub n is equal to s sub n minus s sub n minus one. Or we could write that as equal to this stuff.

So this is n + 1 over n + 10 minus n over n + 9. And this by itself, this is a rule for a sub n. But we could combine these terms, add these two fractions together, and this is actually going to be the case for n greater than one. For n equals 1, s sub one is going to be, well, you can just say a sub one is going to be equal to s sub one.

But then for any other n, we could use this right over here. And if we want to simplify this, well, we can add these two fractions. We can add these two fractions by having a common denominator. So let's see, if we multiply the numerator and denominator here by n + 9, we are going to get so this is equal to n + 1 * n + 9 over n + 10 * n + 9.

And from that, we are going to subtract, let's multiply the numerator and the denominator here by n + 10. So we have n * n + 10 over n + 9 * n + 10.

n + 9 * n + 10, and what does that give us? So let's see, if we simplify up here, we're going to have this is n^2 + 10n + 9, that's that. And then this right over here is n^2 + oh, this is n^2 + 10n, doing that red color, so this is n^2 + 10n.

And remember we're going to subtract this, and so, and we are close to deserving a drum roll. A sub n is going to be equal to our denominator right over here is n + 9 * n + 10, and we're going to subtract the red stuff from the blue stuff.

So you subtract an n from an n squared, those cancel out. Subtract a 10n from a 10n, those cancel out, and you're just left with that blue nine. So there you have it, we've expressed, we've written a rule for a sub n for n greater than one.

More Articles

View All
Seth Klarman's Warning for "The Everything Bubble"
The first thing is, we’ve been in an everything bubble. I think that a lot of money has flowed into virtually everything. You’ve had speculation during that bubble in all kinds of things from crypto to meme stocks to SPACs. That day is Seth Klam, and he …
Scarcity and rivalry | Basic Economic Concepts | Microeconomics | Khan Academy
What we’re going to do in this video is talk about two related ideas that are really the foundations of economics: the idea of scarcity and the idea of rivalry. Now in other videos, we do a deep dive into what scarcity is, but just as a review in everyda…
Partial derivatives and graphs
Hello everyone. So I have here the graph of a two variable function, and I’d like to talk about how you can interpret the partial derivative of that function. So specifically, the function that you’re looking at is f of x, y is equal to x squared times y…
How I Turned $1,500 Into $5.5 Billion
So guys, we’re on our way to Kentucky right now to visit Papa John. And yes, it’s the Papa John, the billionaire Papa John. He’s showing us his house; we’re getting a day in the life, taking you along. And I got a Starbucks, so let’s go! Yeah, about this…
Why Shower Thoughts Are Actually Deep
Everyone loves shower thoughts. It’s the most successful format on this channel. There’s an entire subreddit dedicated to shower thoughts and thousands of TikToks daily talking about profound ideas, paradoxes, and concepts; things that you need to think a…
With Grace | Short Film Showcase | National Geographic
[Music] [Music] Thank you, thank you. [Music] Come on, I’ve been happening. Okay, okay. [Music] You can even take overnight. Sometimes a day can pass or two. Okay. Foreign [Music] Grace, so I went home to catch up some rest. Around 23 hours, I had a knock…