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Aktuel tid:0:00Samlet varighed:10:45

In this video I want to
introduce you to the idea of a geometric sequence. And I have a ton of more
advanced videos on the topic, but it's really a good place to
start, just to understand what we're talking about
when someone tells you a geometric sequence. Now a good starting point is
just, what is a sequence? And a sequence is, you
can imagine, just a progression of numbers. So for example, and this isn't
even a geometric series, if I just said 1, 2, 3, 4, 5. This is a sequence of numbers. It's not a geometric sequence,
but it is a sequence. A geometric sequence is a
special progression, or a special sequence, of numbers,
where each successive number is a fixed multiple of
the number before it. Let me explain what
I'm saying. So let's say my first number is
2 and then I multiply 2 by the number 3. So I multiply it
by 3, I get 6. And then I multiply 6 times the
number 3, and I get 18. Then I multiply 18 times the
number 3, and I get 54. And I just keep going
that way. So I just keep multiplying
by the number 3. So I started, if we want to get
some notation here, this is my first term. We'll call it a1 for
my sequence. And each time I'm multiplying
it by a common number, and that number is often called
the common ratio. So in this case, a1 is equal to
2, and my common ratio is equal to 3. So if someone were to tell
you, hey, you've got a geometric sequence. a1 is equal to 90 and your
common ratio is equal to negative 1/3. That means that the first term
of your sequence is 90. The second term is negative
1/3 times 90. Which is what? That's negative 30, right? 1/3 times 90 is 30, and then you
put the negative number. Then the next number is going
to be 1/3 times this. So negative 1/3 times this. 1/3 times 30 is 10. The negatives cancel out,
so you get positive 10. Then the next number is going
to be 10 times negative 1/3, or negative 10/3. And then the next number is
going to be negative 10/3 times negative 1/3 so it's going
to be positive 10/3. And you could just keep going
on with this sequence. So that's what people talk
about when they mean a geometric sequence. I want to make one little
distinction here. This always used to confuse me
because the terms are used very often in the
same context. These are sequences. These are kind of a progression
of numbers. 2, then 6, then 18, 90, then
negative 30, then 10, then negative 10/3. Then, I'm sorry, this is
positive 10/9, right? Negative 1/3 times negative
10/3, negatives cancel out. Right. 10/9. Don't want to make
a mistake here. These are sequences. You might also see the
word a series. And you might even see
a geometric series. A series, the most conventional
use of the word series, means a sum
of a sequence. So for example, this is
a geometric sequence. A geometric series would be 90
plus negative 30, plus 10, plus negative 10/3, plus 10/9. So a general way to view
it is that a series is the sum of a sequence. I just want to make that clear
because that used to confuse me a lot when I first learned
about these things. But anyway, let's go back to
the notion of a geometric sequence, and actually do a word
problem that deals with one of these. So they're telling us that Anne
goes bungee jumping off of a bridge above water. On the initial jump, the cord
stretches by 120 feet. So on a1, our initial
jump, the cord stretches by 120 feet. We could write it this way. We could write, jump, and then
how much the cord stretches. So on the initial jump,
on jump one, the cord stretches 120 feet. Then it says, on the next
bounce, the stretch is 60% of the original jump, and then
each additional bounce stretches the rope 60% of
the previous stretch. So here, the common ratio, where
each successive term in our sequence is going to be
60% of the previous term. Or it's going to be 0.6 times
the previous term. So on the second jump, we're
going to start 60% of that, or 0.6 times 120. Which is equal to what? That's equal to 72. Then on the third jump, we're
going to stretch 0.6 of 72, or 0.6 times this. So it would be 0.6 times
0.6 times 120. Notice, over here, so on the
fourth jump we're going to have 0.6 times 0.6 times
0.6 times 120. 60% of this jump, so every
time we're 60% of the previous jump. So if we wanted to make a
general formula for this, just based on the way we've defined
it right here. So the stretch on the nth
jump, what would it be? So let's see, we start at 120
times 0.6 to the what? To the n minus 1. How did I get this? Let me write this a
little bit here. So this is equal to 0.6,
actually let me write the 120 first. This is equal to 120
times 0.6 to the n minus 1. How did I get that? Well we're defining the first
jump as stretching 120 feet. So when you put n is equal to 1
here, you get 1 minus 1, 0. So you have 0.6 to the
0th power, and you've just got a 1 here. And that's exactly what happened
on the first jump. Then on the second jump, you put
a 2 minus 1, and notice 2 minus 1 is the first
power, and we have exactly one 0.6 here. So I figured it was n minus 1
because when n is 2, we have one 0.6, when n is 3, we have
two 0.6's multiplied by themselves. When n is 4, we have 0.6
to the third power. So whatever n is, we're taking
0.6 to the n minus 1 power, and of course we're multiplying
that times 120. Now and the question they also
ask us, what will be the rope stretch on the 12th bounce? And over here I'm going
to use the calculator. and actually let me correct
this a little bit. This isn't incorrect, but
they're talking about the bounce, and we could call the
jump the zeroth bounce. Let me change that. This isn't wrong, but I think
this is where they're going with the problem. So you can view the initial
stretch as the zeroth bounce. So instead of labeling it jump,
let me label it bounce. So the initial stretch is the
zeroth bounce, then this would be the first bounce, the second bounce, the third bounce. And then our formula becomes
a lot simpler. Because if you said the stretch
on nth bounce, then the formula just becomes 0.6
to the n times 120, right? On the zeroth bounce, that was
our original stretch, you get 0.6 to the 0, that's
1 times 120. On the first bounce, 0.6 to
the 1, one 0.6 right here. 0.6 times the previous
stretch, or the previous bounce. So this has it in terms of
bounces, which I think is what the questioner wants us to do. So what about the 12th bounce? Using this convention
right there. So if we do the 12th bounce,
let's just get our calculator out. We're going to have 120 times
0.6 to the 12th power. And hopefully we'll get order
of operations right, because exponents take precedence over
multiplication, so it'll just take the 0.6 to the
12th power only. And so this is equal
to 0.26 feet. So after your 12th bounce,
she's going to be barely moving. She's going to be moving about
3 inches on that 12th bounce. Well, hopefully you found
this helpful. And I apologize for the slight
divergence here, but I actually think on some level
that's instructive. Because you always have to make
sure that your n matches well with what your
results are. So when I talked about
your first jump, I said, OK this is 1. And then I had 0.6 to
the zeroth power, so I did n minus 1. But then when I relabeled things
in terms of bounces, this was the zeroth bounce. This makes sense that this is
0.6 to the zeroth power. This is the first bounce,
so this would be 0.6 to the first power. Second bounce, 0.6 to
the second power. It made our equation a
little bit simpler. Anyway, hopefully you found
that Interesting.