Introduction to geometric sequences
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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.