Geometric series word problems: swing | Algebra 2 | Khan Academy

We're told a monkey is swinging from a tree. On the first swing, she passes through an arc of 24 meters. With each swing, she passes through an arc one half the length of the previous swing. So what's going on here?

Let's say this is the top of the rope or the vine that the monkey is swinging from, and so on. That first swing, I could draw a little monkey here, so this is my little monkey. On that first swing, the monkey will go 24 meters. So, might do something like this. Then that arc is 24 meters.

On the second swing, she'd swing back and arc half the length of the previous swing. So, then she would come back, and then it would be half the length. Maybe swing back over here, and then on the next swing, that would be 12. On the next swing, she would swing half of that, which would be six meters. So, she might swing like this, and that makes sense.

That's consistent with our experiences swinging from trees for those of us who've done that. So, let's look at the first choice: which expression gives the total length the monkey swings in her first n swings? So, pause the video and see if you can do that.

You can express it as, actually express it both ways. Express it as a geometric series, but also express it as the sum of a geometric series if we were actually to evaluate it. So let's do this together.

We already said on the first swing, the monkey goes 24 meters. Now on the second swing, and I gave you a hint when I said to express it as a geometric series, she swings half that. Now I could just write a 12 here, but the half is interesting because that's going to be my common ratio for my geometric series.

Every successive swing, the arc length is half the arc length of the last swing. So, it's going to be 24 times one-half, and then on the next swing, it's going to be 24. It's going to be half of this, so it's going to be 24 times one-half times one-half, so that's 24 times one-half to the second power.

This would be the first three swings. Notice that the exponent here we got to the second power. So, the first n swings we are going to get to 24 times one-half, not to the nth power, but to the n minus 1 power. Notice after two swings, we only get to 24 times one-half to the first power, after three swings to the second power.

So after n swings, to the n minus one power. Now, as I said, we don't want to just have this expression; we actually want to know how do we evaluate this. The way we evaluate this is we look at the formula, which we've explained and we've proven in other videos, the formula for a finite geometric series.

So that tells us, and I'll just write over here, the sum of the first n terms is a, where a is the first term. So that's going to be our 24 in this situation. It's a minus a times our common ratio. I already said that our common ratio is one-half to the nth power.

One way I like to remember it is, it is our first term minus the first term that we didn't include, or minus what would have been the term right after this, all of that over 1 minus our common ratio. There are other ways that you might have seen this written; you could factor an a out, and you might have seen something like this: a times 1 minus r to the n, all of that over 1 minus r.

These two are equivalent. But now, let's use this. So this is going to be equal to, actually I'll use this second form right over here. So our first term a is 24. So we're going to have 24 times one minus our common ratio, which is one-half to the nth power.

Well, we're talking about the first n swings, so I'm just going to leave an n right over there, all of that over 1 minus our common ratio, 1 minus one-half. So we could leave it like that or we could simplify it a little bit if we like. One minus one-half is equal to one-half. Twenty-four divided by one-half is equal to 48.

So if you wanted to, you could simplify it to 48 times one minus one-half to the nth power. So either of these would be legitimate. Now the second part, they say, what is the total distance the monkey has traveled when she completes her 25th swing? And they say round your final answer to the nearest meter.

So pause this video and see if you can work that out. All right, well we can just use this expression here. We know that we are completing our 25th swing, so n is 25, and so we'll just put a 25 there.

So that's going to be 48 times 1 minus one-half to the 25th power. Now this is going to be a very, very small, very, very small number. So it's actually going to be pretty close to 48 meters. But let's see what this is equal to, and we're going to round to the nearest meter.

All right, so let's get our calculator out. And so, let's just evaluate one-half. I'll just write that as 0.5 to the 25th power, which, as we said, as we predicted, is a very small number. And then we're going to subtract that from 1, so let's put a negative and then I'll add 1 to it.

That is very close to 1, and so my prediction is holding true. So, if I multiply that times 48, well if we round to the nearest meter, we get back to 48 meters. So, this is going to be 48 meters, and we're done.