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
Give Me Some Slack | Dirty Rotten Survival
This is it. This is the spot, baby! See this slot right here? This is pretty good. So just confirm this is better than an RV ride to a gas station? Yeah, it’s definitely more manly. You better believe it! Right, we’ll get you down there, we’ll get some w…
A Fun, Animated History of the Reformation and the Man Who Started It All | Short Film Showcase
[Music] A most precise and nuanced look into the life of the man, legend, and visionary Martin Luther. One day, when Luther is 21 years old, he experiences something which will affect him for the rest of his life. Suddenly, a thunderstorm—a wild, violent…
Strategies for dividing multiples of 10, 100 and 1000
We’re going to do in this video is get some practice doing division with numbers that are multiples of 10, 100, 1000, things like that. So, let’s say we wanted to compute what 2400 divided by 30 is. Pause this video and see if you can calculate it using w…
THE MAKING OF MY NEW SHOW | BTS for MONEY COURT
This project’s been a year and a half in the making. It’s how long it’s taken. This is one of the most technologically advanced studios in the world. It’s massive; it’s the size of a city block. You know, we’re not saving the world if we met animals in th…
Origins of agriculture | World History | Khan Academy
This timeline here covers 200,000 years, from 200,000 years into the past to the present. Just to get a sense of the scale of this, if we were to go 2,000 years ago to the time of the Roman Empire, that would be roughly here on the timeline. If I were to …
Why Do We Have Two Nostrils?
Hey, Vsauce. Michael here. And our faces have a lot of holes. We have two ears and two eyes. It makes sense, because the difference in time it takes for a sound to reach one ear and the other allows us to localize where the sound is coming from. And havin…