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- 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.