Geometric series with sigma 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 and any nonzero value can be our common ratio can even be a negative value so for example you could have a geometric sequence that looks like this maybe you start at 1 and maybe our common ratio here is what 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 so let's scroll down a little bit so now we're going to talk about geometric geometric series which is really just the sum of a geometric sequence so for example and exam Atrix series would just be a sum of the sequence so if we just said 1 plus we do that same color if we 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 it may be using Sigma notation well we'll start with whatever our first term is and over here if we want to speak in general terms that 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 so 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 a times R squared and then we can keep going plus a times R to the third 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 let's say we get keep going all the way until we get to some a times R to the N a times R to the nth 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 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 I'm going to let write down so this is a times R to the 0 this is a times R to the first R squared our 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 just from K this is a different color from K equals 0 from K equals 0 all the way to K equals n all the way until K equals n of a a the same color of a times R I'm having trouble with colors today a times R to the K power to the K power and so this is using Sigma tation a general way to represent a geometric series where R is some non-zero common ratio it can even be an egg if value