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I'm going to construct a sequence where I start with some number let's say I start with the number a and then each successive term of the sequence I'm going to multiply the pre to get each successive term of the sequence I'm going to multiply the previous term by some fixed nonzero number and I'm going to call that R so the next term is going to be a is going to be a times R and then the term after that I'm going to apply I'm going to multiply this thing times R so it's going to be a if you take if you multiply a R times R that's going to be a our a R squared and then if you were to multiply this term term times R you would get a times R to the third power and you could keep going on and on and on and on and this type of sequence or this type of progression is called a geometric geometric sequence or progression geometric sequence you start with some first value let me circle that in a different color since I already use the green so you start with some first value and then each success to get each successive term you multiply by this fixed number and in this case this fixed number is R and so we call our our common ratio our common ratio why is it called a common ratio well take any two successive terms take this term and this term and divide this term by this term right over here a R to the third divided by a R squared so if you find the ratio between these two things a let me rewrite this same colors so if you took a R to the third power and we're divided by the term before it so if you were divided by a R squared what are you going to be left with well a divided by a is 1 R to the third divided by R squared is just going to be R and this is true if you divide any term by the term before if you find the ratio between any term and the term before it it's going to be R and so that's why it's called a common ratio and so let's get looks a look at some examples of geometric sequences so if I start with the number if I start with the number five so my a is five and then each time I'm going to multiply I'm going to multiply by I don't know a 10 multiplied by 1/7 so then the next term is going to be five over seven five over seven what's the next term going to be I'm going to multiply this thing times 1/7 so that's going to be five seven times 1/7 is 540 ninths so it's going to be 5 over 5 over 7 squared or 49 if I were to multiply this times 1/7 what am I going to get and I'll just change the notation I'll just get 5 times I don't actually know in my head what 7 to the third power is I guess I could calculate it 280 plus 63 let's see so that would be c7 times 40 is 280 7 times 9 is 63 so you're going to get to 343 I believe let me see that I do that right seven times 280 plus 63 343 and you can just keep going so this right over here is an example of a geometric of a geometric sequence started with some first value in each successive value I'm multiplied by 1/7 1/7 is the common ratio here let me give you another one let's say I have this so let's say I have 3 and then let's say I have 6 and let's say I have 12 let's then I have the 24 then I have 48 is this a geometric series and if it is what is the common ratio here well you could figure out the common ratio by just taking any two successive terms and dividing well first you could you could try it with two terms so 12 divided by 6 is 2 so we have to multiply by 2 to go from 12 to 24 you multiplied by 2 to go from 24 to 48 you have to multiply by 2 to go from 3 to 6 we have to multiply by 2 so you had a fixed common ratio for any of these terms we multiplied by 2 and then say multiplied by 3 and and so we didn't multiply by the same thing then it wouldn't be a geometric sequence anymore so this clearly is a geometric sequence I forgot to mention that