Intro to geometric sequences (advanced)

Let's talk about geometric sequences, which is a class of sequences where we start at some number, then each successive number is the previous number multiplied by the same thing. So what am I talking about? Well let's multiply a times r. And then I'm going to get ar. Let's multiply it times, but to get the third term, let's multiply the second term times r. And then what am I going to have? I'm going to have-- it's a different shade of yellow-- I'm going to have ar squared. Multiply by r again, you're going to get ar to the third power, and you just keep on going like that. And this is, the way I've denoted this, this is an infinite geometric sequence. We just keep going on and on and on and on. And the different ways we can denote it, we can denote it explicitly. We could say that our sequence is a sub n starting with the first term going all the way to infinity, with a sub n equaling-- well, we see an a here for any term-- is going to be a times r. And just to be clear, this right over here, a is the same thing as a times r to the zeroth power, r to the 0 is just 1. This second term is ar to the first power. The third term is ar to the third power. It looks like the nth term is going to be ar to the n minus 1 power. So ar to the n minus 1. And you could verify it. If you want the second term, you say a times r to the 2 minus 1, a times r to the first power. It works out. This is defining it explicitly. We could also define it recursively. We could say a sub n from n equals 1 to infinity, with a sub 1 being equal to a. That's the base case. a sub 1 is equal to a, ar to the 0 is just a. Or we could say for n equals 1, and then we could say a-- and I don't even have to really write that because we're making it very clear that a sub 1 is equal to a-- and then we could say a sub n is equal to the previous term, a sub n minus 1, times r, for n is greater than or equal to 2. So this is saying, look, our first term is going to be a, that right over there is a, ar to the 0 is just a, and then each successive term is going to be the previous term times r, which is exactly what we did over there. So let's look at some geometric sequences. So I could have a geometric sequence like this. I could have a sub n, n is equal to 1 to infinity with, let's say, a sub n is equal to, let's say our first term is, I don't know, let's say it is equal to 20. And then r, the number that we're multiplying to get each successive term, let's say it's equal to 1/2. 1/2 to the n minus 1. So what would this sequence actually look like? Well let's think about it. The first term is 20. If you say, if n is 1, this is going to be 1/2 to the 0-th power. So it's going to be 1 times 20. So the first term is 20, and then each time we're multiplying by what? Well here each time we're multiplying by 1/2. So this could be 20 times 1/2 is 10, 10 times 1/2 is 5, 5 times 1/2 is 2.5-- actually let me just write that as a fraction, is 5/2, 5/2 times 1/2 is 5/4, and you can just keep going on and on and on. This is a geometric sequence. Now let me give you another sequence, and tell me if it is geometric. So let's say we start at 1, so then I'm going to go to 2, and then I'm going to go to 6, and then I'm going to go to-- let me see what I want to do-- I want to go to 24. And then I could go to 120, and I go on and on and on. Is this a geometric sequence? Well let's think about what's going on. To go from 1 to 2, I multiplied by 2. To go from 2 to 6, I multiplied by 3. To go from 6 to 24, I multiplied by 4. So I'm always multiplying not by the same amount. You have to multiply by the same amount in order for it to be a geometric sequence. Here I'm multiplying it by a different amount. So this sequence that I just constructed has the form, I have my first term, and then my second term is going to be 2 times my first term, and then my third one is going to be 3 times my second term, so 3 times 2 times a. My fourth one is 4 times the third term, so 4 times 3 times 2 times a. And we go on and on and on. So this sequence, which is not a geometric sequence, we can still define it explicitly. We could say that its set or it's the sequence a sub n from n equals 1 to infinity with a sub n being equal to, let's see the fourth one is essentially 4 factorial times a. Well, actually, if we look at this particular, these particular numbers our a is 1. So this is actually, let me write this, this is 1, this is 2 times 1, this is 3 times 2 times 1, this is 4 times 3 times 2 times 1. And so a sub n is just equal to n factorial. This right over here, which is not a geometric sequence, describes exactly this sequence right over here. Just to get some practice with-- Here we've defined it explicitly, but we can also define it recursively. We could also say-- do it in white-- we could also say that a sub n takes us from n equals 1 to infinity, with a sub 1, or maybe at a sub 1 is equal to 1. That's our first term. And then each successive term is going to be equal to the previous term times n. So the second term is equal to the previous term times 2. The nth term is going to be the previous turn times n So this is another valid way of defining it.