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## AP®︎/College Calculus BC

### Unit 10: Lesson 2

Working with geometric series- Worked example: convergent geometric series
- Worked example: divergent geometric series
- Infinite geometric series
- Infinite geometric series word problem: bouncing ball
- Infinite geometric series word problem: repeating decimal
- Proof of infinite geometric series formula

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# Infinite geometric series word problem: bouncing ball

AP.CALC:

LIM‑7 (EU)

, LIM‑7.A (LO)

, LIM‑7.A.3 (EK)

, LIM‑7.A.4 (EK)

Watch Sal determine the total vertical distance a bouncing ball moves using an infinite geometric series. Created by Sal Khan.

## Video transcript

Let's say that we have
a ball that we dropped from a height of 10
meters, and every time it bounces it goes half as
high as the previous bounce. So for example, you
drop it from 10 meters. The next time its peak height
is going to be at 5 meters. So the next time around,
on the next bounce, let me draw in that
same orange color. And the next bounce the ball
is going to go 5 meters. This distance right over
here is going to be 5 meters. And then the bounce after that
is going to be half as high. So it's going to go
2 and 1/2 meters. And it's just going
to keep doing that. So it's going to go 2 and
1/2 meters right over here. And what I want to think
about in this video is what is the total vertical
distance that the ball travels? So let's think about
that a little bit. So it's first going to travel
10 meters straight down. So it's going to travel
10 meters just like that, and then it's going to travel
half of 10 meters twice. It's going to go up 5
meters, up half of 10 meters, and then down half of 10 meters. Let me put it this way. So each of these is
going to be 10 meters. Actually, I don't have
to write the units here. Let me take the
units out of the way. Let me write that clear. So the first bounce,
once again, it goes straight down 10 meters. Then on the next bounce it's
going to go up 10 times 1/2. And then it's going to
go down 10 times 1/2. Notice we just care about
the total vertical distance. We don't care about
the direction. So it's going to go up 10
times 1/2, up 5 meters, and then it's going
to go down 5 meters. So it's going to travel a total
vertical distance of 10 meters, 5 up and 5 down. Now what about on this jump, or
on this bounce, I should say. Well here it's going to go
half as far as it went there. So it's going to go 10 times 1/2
squared up, and then 10 times 1/2 squared down. And I think you
see a pattern here. This looks an awful lot
like a geometric series, an infinite geometric series. It's going to just keep on going
like that forever and ever. So let's try to clean this
up a little bit so it looks a little bit more like a
traditional geometric series. So if we were to simplify
this a little bit we could rewrite
this as 10 plus 20. 20 times 1/2 to the first power,
plus 10 1/2 times 1/2 squared plus 10 times 1/2 squared
is going to be 20 times 1/2 squared, and we'll just
keep on going on and on. So this would be a
little bit clearer if this were a 20
right over here. But we could do that. We could write 10 as
negative 10 plus 20, and then we have plus all of
this stuff right over here. Let me just copy and paste that. So plus all of this
right over here. And we can even
write this first. We can even write this 20
right over here is 20 times 1/2 to the 0 power
plus all of this. So now it very
clearly looks like an infinite geometric series. We can write our entire
sum, and maybe I'll write it up here since I don't
want to lose the diagram. We could write it
as negative 10. That's that negative
10 right over here. Plus the sum from
k is equal to 0 to infinity of 20 times our
common ratio to the k-th power. So what's this going to be? What's this going
to turn out to be? Well we've already
derived in multiple videos already here that the sum of an
infinite geometric series, so the sum from k equals 0
to infinity of a times r to the k is equal
to a over 1 minus r. So we just apply
that right over here. This business right
over here is going to be equal to 20
over 1 minus 1/2, which is the same
thing as 20 over 1/2, which is the same thing
as 20 times 2, or 40. So what's the total vertical
distance that our ball travels? It's going to be negative
10 plus 40, which is equal to 30 meters. Our total vertical distance that
the ball travels is 30 meters.