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### Course: Get ready for Precalculus > Unit 6

Lesson 6: Geometric series# Geometric series word problems: swing

The length a monkey passes when swinging from a tree can be modeled by a geometric series! We learn that each swing's length is half the previous one, forming a finite geometric series. By applying the sum formula, we calculate the total distance the monkey swings.

## Want to join the conversation?

- if you swing a pendulum, and each time it swings 1/2 of the distance, would it go on forever?

Thanks in advance!(34 votes)- Yes - theoretically, it would never reach zero. It would technically just
*approach*zero.(35 votes)

- That is a cockroach Sal not a monkey :D(27 votes)
- Physically, isn't 0.5 of the distance of the previous swing impossible ?(9 votes)
- I think so too. 1/2 the distance of the previous swing means the pendulum would come to a stop exactly at the
**bottom of the swing**. You'd only get that if you move a wall at that position to crash into after the first swing, or if you have some external force acting on the pendulum.(8 votes)

- please help me with this question:- A swinging pendulum covers 32 centimetres in the first swing, 24 centimetres in the second swing and 18 centimetres in the third and so on. what is the total distance it swings before coming to rest?(4 votes)
- You can create a geometric series based on this. a1 is 32 centimeters, r = 0.75. Now you need to sum the series up. I believe that you need to treat this as an infinite geometric series and use the formula for an infinite geometric series. a1/(1-r). So plug in the values and see what the answer is. This is my understanding and I hope this helps you!(4 votes)

- What if we were given the final distance the monkey traveled but we weren't given the nth term. How would we solve for n?(2 votes)
- you could possibly use the sum formula and start from there(4 votes)

- please help! the first question asks the total length of the swing after the nth swing right? so that should be a very small length as the swing length continuously decreases.

and in the second question, the distance after her 25h swing, the answer is 48m, not a really small number. so...

in that case, we should not use that formula to answer the first question? or the first question should be something like the total length of all the swings added?(2 votes)- Total length (or total distance) is already saying total length of all swings added. if it just wanted the length of the 25th swing it would ask something like "what is the length of the 25th swing". Saying total length is saying it wants all swings added up.(2 votes)

- Can anyone help me understand why at the last step, which (forgive me, I can't type fractions here) is 48(1 - 1/2 ^25), you can't first distribute the 48 to both terms inside the parentheses and get an equivalent answer to the one you get when you evaluate it as is? I know we do computation inside parentheses first, I'm just wondering why that doesn't work. Or am I just entering it into my calculator wrong? Or does it have something to do with that fraction raised to a power?(2 votes)
- We can distribute the 48, but because of the exponent it's hardly worth the trouble.

48(1 − (1∕2)^25)

= 48 − 48 ∙ (1∕2)^25

= 48 − 3 ∙ 2^4 ∙ (1∕2)^4 ∙ (1∕2)^21

= 48 − 3 ∙ 2^4 ∙ 2^(−4) ∙ (1∕2)^21

= 48 − 3 ∙ (1∕2)^21(2 votes)

- in a-ar^2, why are we subracting "what would be the first term after n-1?(2 votes)
- Am I the only one that's been doing this for 3 hours and still doesn't understand(1 vote)
- In this problem, a monkey initially swings for 24m. Then on the next pass, it swings half the first length, which means first swing length * 0.5, i.e. 12m. After that it continues to decrease by half.

So on the third pass, we get

12 * 0.5 = 6m

On the fourth pass, we get

6 * 0.5 = 3m

And so on.

Now notice on the second pass, we get

24 * 0.5

On the third pass, we get

(24 * 0.5) * 0.5

= 24 * 0.5²

On the fourth pass, we get

(24 * 0.5 * 0.5) * 0.5

= 24 * 0.5³

The power keep increasing.

So we can claim, on the n-th pass, the power is (n - 1), and the length on the n-th pass is 24 * 0.5ⁿ⁻¹.

A few lessons ago, we've learnt the Finite Geometric series formula`S = [a * (rⁿ - 1)] / (r - 1)`

where a is the first term, r is the common ratio and n is the number of terms.

Since we are required to represent S in terms of n, we now only have to find a and r.

a = 24m, which is provided in the problem statement.

r = 0.5, which we observed previously.

Therefore,`S = 24 * (0.5ⁿ - 1) / (0.5 - 1)`

S = 48 * (1 - 0.5ⁿ)

To find out the total distance the monkey travelled on the 25th swing, simply substitute n by 25 and you get your answer.(2 votes)

- Does this mean her total length of swings never exceed than 48 meters? but approaches 48 infinitely?(1 vote)
- I think it's because you can actually distribute the 24+24*1/2+24*(1/2)^2+...+24*(1/2)^n-1 into 24*(1+1/2+1/4+1/8+1/16+...+1/2^n) which will be infinitely close to 24*2 yet not ever equal to 24*2 which is 48.

Hope this helps.(1 vote)

## Video transcript

- [Instructor] 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 1/2 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 like to draw a little monkey here, so this is my little monkey. So on the first swing, the
monkey will go 24 meters. Might do something like this. Then that arc is 24 meters, and then on the second swing, it would be, she'd swing back at an arc half the length of the previous swing. So then she would come back and then it would be half the length, and so maybe swing back over here. And then on the next, so that'd be 12, and then on the next swing,
she would swing half of that, which would be six meters. And so she might swing like this, and that makes sense. That's consistent with our
experiences swinging from trees, for those of us who have done that. (laughs) 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, and you can express it as, actually express it two ways, express it as a geometric series, but also express it as the
sum of a geometric series if it were actually evaluated. So let's do this together. So 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 1/2 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 1/2 times 1/2. So that's 24 times 1/2
to the second power. And so 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 1/2, not to the nth power, but to the n minus one power. Notice, after two swings, we only get to 24 times 1/2 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 wanna
just have this expression. We actually wanna know,
how do we evaluate this? And 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 it over here, the sum of 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 1/2, to the nth power. So 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've been
the term right after this. All of that over one
minus our common ratio. And there's other ways that
you might've seen this written. You could factor an a out, and you might have seen
something like this: a times one minus r to the n, all of that over one 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 1/2, to the nth power, well we're talking about the first n swing, so I'm
just going to leave an n right over there. All of that over one
minus our common ratio, one minus 1/2. So we could leave it like that or we could simplify it
a little bit if we like. One minus 1/2 is equal to 1/2, 24 divided by 1/2 is equal to 48, so if you wanted to, you could simplify it to 48 times one minus 1/2 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. And 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 one minus 1/2 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 1/2, 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 one, so I'll just put a negative
and then I'll add one to it. And so that is very close to one. 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.