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
2015 AP Calculus AB/BC 1ab | AP Calculus AB solved exams | AP Calculus AB | Khan Academy
The rate at which rainwater flows into a drain pipe is modeled by the function R, where R of T is equal to 20 sin of T^2 over 35 cubic feet per hour. T is measured in hours, and 0 is less than or equal to T which is less than or equal to 8. So T is going …
I Lost. My Response To Boxing Michael Reeves | Creator Clash
All right, hey guys. So this is going to be one of the very few unscripted videos I ever do on this channel. But I feel like it’s about time that I address a lot of videos that have been going around of me boxing Michael Reeves, me losing. And I’ve notice…
When you stop trying, it happens | The psychology of the flow state
We often hear of remarkable people who, through dedication and practice, seem to become one with their craft. An example of such a person is Tsao-fu, a character from Taoist literature who wished to become a skilled charioteer. So, he seized the opportuni…
More Questions Than Answers | LA 92
I think it is devastating to the image of this city and especially to our police department. JOHN MACK: It’s very apparent that some– not all, but some– of those officers are clearly out of control. And they have to ultimately be willing to take a good, …
I Spent $100,000 On A Stock Picking Monkey
What’s up you guys? It’s Graham here. So if you thought this year cannot get any more unpredictable, it just did because I have a hundred thousand dollars on the line for the sole purpose of testing an odd yet unique experiment to see whether or not a mon…
What is Morality?
If I steal from the rich and feed to the poor, is that good or bad? If I drive over the speed limit to get my sick child in the hospital, is that good or is that bad? What is good and what is bad? What is morality, and do you as a person have morals? Mor…