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

Worked examples: Summation notation | Accumulation and Riemann sums | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

We're told to consider the sum 2 plus 5 plus 8 plus 11. Which expression is equal to the sum above? And they tell us to choose all answers that apply. So, like always, pause the video and see if you can work through this on your own.

When you look at the sum, it's clear you're starting at two, you're adding three each time, and we also are dealing with four total terms. Now, we could try to construct an equation here or an expression using sigma notation, but instead what I like to do is look at our options. We really only have to look at these two options here and expand them out. What sum would each of these be?

Well, this is going to be the same thing as we're starting at n equals 1. So this is going to be 3 times when n equals 1, 3 times 1 minus 1, and then plus... then we'll go to n equals 2, 3 times 2 minus 1. Then we're going to go to n equals 3, so plus 3 times 3 minus 1. And then finally we're going to go to n equals 4, so plus 3 times when n equals 4. This is 3 times 4 minus 1.

So just to be clear, this is what we did when n equals 1. Let me write it down: n equals 1, this is what we got; when n equals 2, this is what we got; when n equals 3, and this is what we got when n is equal to four. We stopped at n equals four because that tells us right over there, we start at n equals one and we go all the way to n equals four.

So what does this equal? Let's see. 3 times 1 minus 1 is 2, so this is looking good so far. 3 times 2 minus 1, that's 6 minus 1, that is 5, so still looking good. 3 times 3 minus one, that's nine minus one; once again, still looking good. Three times four minus one, that is eleven. So we like this choice; I would definitely select this one.

Now, let's do the same thing over here. When n is equal to 0, it's going to be 2 plus 3 times 0, so that's just going to be 2. And then plus, when n is equal to 1, it's going to be 2 plus 3 times 1, which is 5. This is starting to look good now. When n is equal to... this was n equals 0, this was n equals 1, so now we're at n equals 2. So at n equals 2, 2 plus 3 times 2 is 2 plus 6, which is 8.

And this makes sense; every time we increase n by 1, we are adding another 3, which is consistent with what we saw there. We start at 2 when n equals 0; this is just that 3n is just 0. So you start at 2, and then every time you increase n by 1, you keep adding 3 again. Finally, when n is equal to 3, 2 plus 3 times 3 is 2 plus... or is 11, I should say. And so this also is exactly the same sum.

So I feel good about both of these. Let's do one more example. So we're here, we're given the sum, and we're saying choose one answer: which of these is equivalent to this sum right over here? Well, like we did before, let's just expand it out. And what's different here is that we just have a variable, but that shouldn't make it too much more difficult.

So let's do the situation, and I'll write it out. Let's do the situation when n is equal to one. When n is equal to one, it's going to be k over 1 plus 1, k over 1 plus 1. And I'll write it out; this is n is equal to 1. Then plus, when n is equal to 2, it's going to be k over 2 plus 1. This is n is equal to 2.

And then we're going to keep going. When n is equal to 3, it's going to be k over 3 plus 1, that's n is equal to 3. And then plus, finally, because we stop right over here at n equals 4, when n equals 4, it's going to be k over 4 plus 1; that's when n is equal to 4.

So this is all going to be equal; I'll just write it over here. This is equal to k over 2 plus k over 3 plus k over 4 plus k over 5, which is exactly this choice right over here. Actually, if I had looked at the choices ahead of time, I might have even been able to save even more time by just saying, well, look, actually, if you just try to compute the first term when n is equal to 1, it would be k over 2.

Well, only this one is starting with a k over 2. This one has no k's here, which is sketchy. They're trying to look at the error where you try to replace the k with the number as well, not just the n. So that's what they're trying to do here. Here they're trying to... let's see, well this... this isn't as obvious what they're even trying to do.

Right over here, where they put the k in the denominator, and here if you swapped the n and the k's, then you would have gotten this thing right over here. So we definitely feel good about choice A.

More Articles

View All
Probability with combinations example: choosing groups | Probability & combinatorics
We’re told that Kyra works on a team of 13 total people. Her manager is randomly selecting three members from her team to represent the company at a conference. What is the probability that Kyra is chosen for the conference? Pause this video and see if yo…
The Mexican-American War | AP US History | Khan Academy
This is a painting of U.S. General Winfield Scott entering Mexico City on September 15, 1847. Scott landed with a U.S. naval fleet several weeks beforehand. He bombarded the coastal stronghold of Veracruz and then fought his way inland toward the capital.…
Peter Lynch: How to Invest in 2023
Peter Lynch: The man, the myth, the legend. He ran the Magellan fund at Fidelity between 1977 and 1990, where he achieved a 29.2 percent annual return. The guy is an investing master. He also wrote the book “One Up On Wall Street,” which you know at this …
Monarch Butterflies Get Tiny Radio Trackers | Expedition Raw
[Music] He’s like a little kid. It’s wonderful. We’re trying to put the first electronic tag on a free flying migrating monarch butterfly. If that works, then we could for the first time really follow them in the wild, how they migrate, and find out exact…
Clarifying standard form rules
We’ve talked about the idea of standard form of a linear equation in other videos, and the point of this video is to clarify something and resolve some differences that you might see in different classes in terms of what standard form is. So everyone agr…
Multiplying and dividing by powers of 10
In another video, we introduce ourselves to the idea of powers of 10. We saw that if I were to just say 10 to the first power, that means that we’re just really just going to take 1. If we have 10 to the second power, that means that we’re going to take …