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

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.