Intro to geometric sequences

in this video I want to introduce you to the idea of a geometric sequence geometric sequence and I have a ton of more advanced videos on the topic but it's really a good place to start just understand what we're talking about when someone tells you a geometric sequence now a good starting point is just what is a sequence and the sequence is you can imagine just a progression of numbers so for example and this isn't even a geometric series if I just said you know 1 comma 2 comma 3 comma 4 comma 5 this is a sequence of numbers it's not a geometric sequence but it is a sequence now a geometric sequence is a special progression or a special sequence of numbers where each each successive number is a is a fixed multiple of the number before it so let me explain what I'm saying so let's say my first number is 2 and then I multiply 2 by the number 3 so I multiply it by 3 I get 6 and then I multiply 6 times the number 3 and I get 18 and then I multiply 18 times the number 3 and I get 54 and I just keep going that way so I just keep multiplying by the number 3 so I started if we want to have get some notation here this is my first term we'll call it a1 for my sequence and each time I'm multiplying it by a common number and that number is often called the common ratio so in this case a 1 is equal to 2 and my common ratio is equal to 3 common common ratio so if someone were to tell you hey you've got a geometric sequence a1 is equal to 90 and your common ratio is equal to negative 1/3 that means that the first term of your sequence is 90 the second term is negative 1/3 times 90 which is what that's negative thirty right one third times ninety is thirty and then you put the negative number then the next number is going to be one third times this so one-third or negative one third times this one third times 30 is ten the negatives cancel out so you get positive ten then the next number is going to be ten times negative one third or negative ten thirds and then the next number is going to be negative ten thirds times negative one thirds it's going to be positive ten thirds and you could just keep going on with this sequence so that's what people talk about when they mean a geometric sequence and I want to make one little distinction here this always used to confuse me because the terms are used very often in the same context these are sequences these are kind of a progression of numbers to than 6 and 1890 the negative 30 the 10 the negative 10 over 3 then sorry then this is positive 10 over 9 right negative 1/3 times negative 10 over 3 negatives cancel out all right 10 over 9 I want to make a mistake here these are sequences you might also see the word a series you might also see the word a series and you might even see a geometric geometric series a series the most conventional use of the word series means a sum of a sequence so for example this is a geometric sequence a geometric series would be 90 plus negative 30 plus 10 plus negative 10 over 3 plus 10 over 9 so a general way to view it is that a series is the sum of a sequence I always just want to make that clear because that used to confuse me a lot when I first learned about these things but anyway let's go back to the notion of a geometric sequence and actually do a word problem that deals with one of these so they tell us they're telling us that an goes bungee jumping off of a bridge above water so an and bungee jumping bungee and is bungee jumping on the initial jump the cord stretches by 120 feet so on a 1-hour original initial jump the cord stretches by 120 feet we could write it this way let's write it this way we could write jump and then how much the cord stretches stretches so on the first on the initial jump on jump one the cord stretches 120 feet then it says on the next bounce the stretch is 60 percent of the original jump and then each additional bounce stretches the rope 60 percent of the previous stretch so here the common ratio where each successive term in our sequence is going to be 60 percent of the previous term where it's going to be 0.6 times the previous term so on the second jump we're going to start 60 percent of that or point 0.6 times 120 which is equal to what that's equal to 72 then on the third jump we're going to stretch Oh point 6 of 72 or 0.6 times this so it'd be 0.6 times 0.6 times 120 and then on the fourth jump on the fourth jump notice over here so on the fourth jump we're going to have zero point six times zero point six times zero point six times 120 zero 60 percent of this jump so every time we're 60 percent of the previous jump so if we wanted to make a general formula for this just based on the way we've defined it right here so the stretch on the nth jump stretch on nth jump what would it be so let's see we start at 120 120 times times 0.6 times 0.6 let me do the point six here times zero point six to the what to the N - one now how did I get this let me write this a little bit near so this is equal to zero point six actually let me write the 121st this is equal to 120 times zero point six to the n minus one how did I get that well we're defining the first jump as stretching 120 feet so when you put n is equal to one here you get one minus one zero so you have point six to the 0th power so you just get a 1 here and that's exactly what happened on the first jump then on the second jump you put a 2 minus 1 and notice 2 minus 1 is the first power and we have exactly one point six here so I figured it was n minus 1 because when n is 2 we have one point six when n is 3 we have two point six is multiplied by themselves when n is 4 we have point six to the third power so whatever n is we're taking point six to the n minus one power and of course we're multiplying that times 120 now in the question they also ask us what will be the Rope stretch on the twelfth on the twelfth bounce on the twelfth bounce and over here over here I'm going to I'm going to use the calculator and actually actually let me let me correct this a little bit this isn't incorrect but they're talking about the bounce and we could call the jump the zeroth bounce so let me change that this isn't wrong but I want to make it a little bit more I think this is where they're going with the problem so you can view the initial strength stretch as the zeroth bounce so let me instead of labeling it jump let me label it bounce so the initial stretch is the zeroth bounce then this would be the first bounce the second bounce the third bounce and then our formula becomes a lot simpler because if you said the stretch stretch on n bounce then the formula just becomes 0.6 to the N times 120 right on the zeroth bounce that was our original stretch you get 0.6 to the 0 that's 1 times 120 on the first bounce 0.6 to the 1.6 to the one you have one point six right here point six times the previous stretch or the previous bounce so this has it in terms of bounces which i think is what the questioner wants us to do so what about the twelfth bounce twelfth bounce using this convention right there so if we do the twelfth bounce let's just get our calculator out we're going to have 120 120 times 0.6 to the to the twelfth power and it hopefully will get order of operations right because exponents take precedence over multiplication so it'll just take the point six to the twelfth power only and so this is equal to point two six feet so this is equal to point or 0.26 feet so after your twelfth bounce she's going to be barely moving she's going to be moving about three inches on that twelfth bounce well anyway hopefully you found this helpful and I apologize for the slight divergence here but I actually don't think on some level that's instructive because you always have to make sure that your n matches well with what your results are so when I talked about just you know your first jump I said okay this is one and then I had a point six to the 0th power so I did n minus one but then when I relabeled things in terms of bounces this was the zeroth bounce then this makes sense so this is 0.6 to the zeroth power this is the first bounce then this would be 0.6 to the first power second bounce 0.6 to the second power and made our equation a little bit simpler anyway hopefully you found that interesting