Geometric series (with summation notation)
In the last video we saw that a geometric progression, or a geometric sequence, is just a sequence where each successive term is the previous term multiplied by a fixed value. And we call that fixed value the common ratio. So, for example, in this sequence right over here, each term is the previous term multiplied by 2. So 2 is our common ratio. And any non-zero value can be our common ratio. It can even be a negative value. So, for example, you could have a geometric sequence that looks like this. Maybe start at one, and maybe our common ratio, let's say it's negative 3. So 1 times negative 3 is negative 3. Negative 3 times negative 3 is positive 9. Positive 9 times negative 3 is negative 27. And then negative 27 times negative 3 is positive 81. And you could keep going on and on and on. What I now want to focus on in this video is the sum of a geometric progression or a geometric sequence, and we would call that a geometric series. Let's scroll down a little bit. So now we're going to talk about geometric series, which is really just the sum of a geometric sequence. So, for example, a geometric series would just be a sum of this sequence. So if we just said 1 plus negative 3, plus 9, plus negative 27, plus 81, and we were to go on, and on, and on, this would be a geometric series. And we could do it with this one up here just to really make it clear of what we're doing. So if we said 3 plus 6, plus 12, plus 24, plus 48, this once again is a geometric series, just the sum of a geometric sequence or a geometric progression. So how would we represent this in general terms and maybe using sigma notation? Well, we'll start with whatever our first term is. And over here if we want to speak in general terms we could call that a, our first term. So we'll start with our first term, a, and then each successive term that we're going to add is going to be a times our common ratio. And we'll call that common ratio r. So the second term is a times r. Then the third term, we're just going to multiply this one times r. So it's going to be a times r squared. And then we can keep going, plus a times r to the third power. And let's say we're going to do a finite geometric series. So we're not going to just keep on going forever. Let's say we keep going all the way until we get to some a times r to the n. a times r to the n-th power. So how can we represent this with sigma notation? And I encourage you to pause the video and try it on your own. Well, we could think about it this way. And I'll give you a little hint. You could view this term right over here as a times r to the 0. And let me write it down. This is a times r to the 0. This is a times r to the first, r squared, r third, and now the pattern might be emerging for you. So we can write this as the sum, so capital sigma right over here. We can start our index at 0. So we could say from k equals 0 all the way to k equals n of a times r to the k-th power. And so this is, using sigma notation, a general way to represent a geometric series where r is some non-zero common ratio. It can even be a negative value.