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- [Instructor] We're told consider the sum two plus five plus eight plus 11. Which expression is equal to the sum above? And they tell us choose all answers that apply. So like always, pause the video, and see if you can work through this on your own. So 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 you have 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 one. So this is going to be three times, when n equals one, three times one minus one. And then plus, then we'll go to n equals two. Three times two minus one. Then we're gonna go to n equals three. So plus three times three minus one. And then finally, we're gonna go to n equals four. So plus three times, when n equals four, this is three times four minus one. So just to be clear, this is what we did when n equals one. Let me write it down. N equals one. This is what we got when n equals two. This is what we got when n equals three. And this is what we got when n is equal to four. And we stopped at n equals four 'cause it tells us right over there, we start at n equals one and we go all of the way to n equals four. So what does this equal? So let's see, three times one minus one is two. So this is looking good so far. Three times two minus one, that's six minus one. That is five. So still looking good. Three times three minus one, that's nine minus one. Once again, still looking good. Three times four minus one, that is 11. So we like this choice, so I would definitely select this one. Now let's do the same thing over here. When n is equal to zero, it's going to be two plus three times zero. So that's just going to be two. And then plus, when n is equal to one, it's going to be two plus three times one, which is five. This is starting to look good. Now when n is equal to, this was n equals zero, this was n equals one, so now we're at n equals two. So at n equals two, two plus three times two is two plus six, which is eight. And this makes sense. Every time we increase n by one, we are adding another three, which is consistent with what we saw there. We start at two, when n equals zero, this is just, that three n is just zero. So you start at two, and then you keep, every time you increase n by one, you are just adding three again. So finally, when n is equal to three, two plus three times three is two 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 gonna be k over one plus one. K over one plus one, and I'll write it out. This is n is equal to one. And then plus, when n is equal to two, it's gonna be k over two plus one. This is n is equal to two. And then we're gonna keep going. When n is equal to three, it's gonna be k over three plus one. That's n is equal to three. And then plus, finally, 'cause we stop right over here at n equals four, when n equals four, it's going to be k over four plus one. That's when n is equal to four. So this is all going to be equal, I'll just write it over here, this is equal to k over two, plus k over three, plus k over four, plus k over five, which is exactly this choice right over, right over here. And 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 one, it would be k over two. Well, only this one is starting with a k over two. This, 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 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've gotten this thing right over here. So we definitely feel good about choice A.
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