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
DoorDash at YC Summer 2013 Demo Day
Hi, we’re DoorDash, and we enable every restaurant to deliver for customers. We offer restaurant food delivery in under 45 minutes, and for restaurant owners, we provide our own drivers and manage the logistics of delivery. Now, you might think that food…
Why Banks Are Collapsing (DO THIS ASAP)
Bank Regulators have seen Silicon Valley Bank in the largest bank failure since the Great Recession. Customers were rushing to take their money out. There are recent developments that concern a few Banks. “What’s up guys, it’s Graham here, and in the las…
15 Ways to Make $10,000 Per Month as a Student
As a student, you typically got no time, no capital, and no experience, right? So how do we know it’s possible for you to make $110,000 a month despite all of these obstacles? Well, because we’ve done it, and we’ve seen thousands of other people do it too…
Derivatives of sin(x) and cos(x) | Derivative rules | AP Calculus AB | Khan Academy
What I’d like to do in this video is get an intuitive sense for what the derivative with respect to x of sine of x is and what the derivative with respect to x of cosine of x is. I’ve graphed y is equal to cosine of x in blue and y is equal to sine of x i…
Jeff Bezos: The electricity metaphor
When you think about resilience and technology, it’s actually much easier. You’re going to see some other speakers today, I already know, who are going to talk about breaking-bones stuff, and, of course, with technology it never is. So it’s very easy, com…
The Key To A Long-term Relationship
How do you know when you meet your perfect partner that you think is going to be with you for life? Maybe you get married, maybe you don’t. But my point is, why are they the right one for you? You have to think long term. Are they your partner because th…