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let's talk about geometric geometric sequences 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 I'm going to get AR let's multiply it times 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 a it's a different shade of yellow I'm going to have a R squared multiplied by R again you're going to get a R 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 with 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 with a sub n equaling well we see an a here for any term it's 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 0th power R to the 0 is just 1 this this second term is a R to the first power the third term is AR to the third power it looks like the nth term is going to be a R to the n minus 1 power so a R to the n minus 1 and you could verify it if you want the second term you can 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 with a sub 1 being equal to a that's the base case a sub 1 is equal to a a R to the 0 is just a and 4 we could say for for N equals 1 and then we could say a and I don't even have to really write that because we're making 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 are 4 n is greater than or equal to 2 so this is saying ìlook our first term is going to be a that right over there is a a R 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 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 our 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 the sequence actually look like well let's think about it the first term is 20 if you say if you n is 1 this is going to be 1/2 to the 0th power so it's just 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 it's going to be 20 times 1/2 is 10 10 times 1/2 is 5 5 times 1/2 is 2.5 actually just write that as a fraction it's 5 halves 5 halves 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 I started one and so then I'm going to go to two 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 I multiplied by 2 to go from 2 to 6 I multiplied by 3 to go from 6 to 24 I multiplied I multiplied by 4 so I'm always multiple 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 that I just constructed has the form I have my first term and then my second term is going to be two times two times my first term and then my third one is going to be third taught three times my second term so three times two times ay my fourth one is four 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 it's this set or the 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 what if we look at this particular these particular numbers are 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 8 a sub n is just equal to is just equal to n n factorial this this right over here which is not a geometric sequence describes exactly this sequence right over here just to get some practice with Ricardo here we've defined it explicitly but we could also define it recursively we could also say do it in white we could also say that a sub n 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 to the nth term is going to be the previous term times n so this is another valid way valid way of finding it