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Current time:0:00Total duration:6:17

Geometric series introduction


Video transcript

in this video we're going to study geometric series and to understand that I'm going to construct a little bit of a table to understand how our money could grow if we keep depositing let's say a thousand dollars a year in a bank account so let's say this is the year and we're going to think about how much we have at the beginning of the year and then this is the dollars in our account and let's say that the bank is always willing to give us five percent per year which is pretty good it's very hard to find a bank account that will actually give you five percent growth per year so that means if you put $100 in at the end of a year or exactly a year later be one hundred five dollars if you put a thousand dollars in a year later it'd be a thousand fifty would be five percent larger and so let's say that we want to put a thousand dollars in per year and I want to think about well what is going to be my balance at the beginning of year 1 at the beginning of year 2 at the beginning of year 3 and then see if we can come up with a general expression for the beginning of year n so year 1 right at the beginning of the year I put in 1,000 dollars in the account that's pretty straightforward but then what happens in year 2 I'm going to deposit a thousand dollars but then that original thousand dollars that I have would have grown so I'm going to deposit $1,000 and then the original thousand dollars that I put at the beginning of year one that is now grown by five percent growing by five percent is the same thing as multiplying by 1.05 so this is now going to be plus 1,000 dollars times 1.05 fairly straightforward now what about the beginning of year 3 how much would I have in the bank account right when I made that first that year 3 deposit pause this video see if you can figure that out well just like at the beginning of year 2 and the beginning of year 1 we're going to make a thousand dollar deposit but now the money from year two has grown by 5% so this is now going to be one thousand dollars times 1.05 and then that money that we originally deposited from year one that was one thousand times 1.05 and year two that's going to grow by another five percent and so this is going to be plus one thousand times 1.05 times 1.05 we're going by another five percent well we could just rewrite this part right over here as 1.0 five squared so do you see a general pattern that's going to happen here well as you go to year and in fact pause the video again and see if you could write a general expression you're gonna have to do a little bit of this dot dot dot action in order to do it but see if you could write a general expression for a year and well for year n you're going to make that original thousand dollars at the beginning of year n and then you're going to have one thousand one thousand times 1.05 for that thousand dollars that you deposit at the beginning of year n minus one and then this is just going to keep going and it's going to go all the way to plus one thousand dollars two times 1.05 ^ the number of years you've been compounding so you could view this thousand dollars as the one that you put in year one and then how many years has it compounded well when you go from one to two you've compounded one year when you go from one to three you've compounded two years so when we're talking about the beginning of year n you go up to the exponent that is one less than that and so this is going to be to the N minus one power so what we just did here is we've just constructed each one of these when we're saying okay how much money do we have in our bank account at the beginning of year three or how much do we have in our bank account at the beginning of year ñ these are geometric series and I'll write that word down gyeo geometric series now just as a little bit of a review or it might not be review might be a primer series are related to sequences and you can really view series as sums of sequel is sequences and let me go down a little bit so that you can so we have a little bit more space a sequence is an ordered list of numbers a sequence might be something like well let's say an or we have a geometric sequence and a geometric sequence each successive term is the previous term times a fixed number so let's say we start at 2 and every time we multiply by 3 so we'll go from 2 2 times 3 is 6 6 times 3 is 18 18 times 3 is 54 this is a geometric sequence ordered list of numbers now if we want to think about the geometric series or the the one that's analogous to this is that we would sum the terms here so this would be 2 plus 6 plus 18 plus 54 or we could even write it and this will look similar to what we had just done with our little savings example is this is 2 plus 2 times 3 plus 2 times 3 squared plus 2 times 3 to the third power and so with the geometric series you're going to have a sum where each successive term in the expression is equal to if you put them all in order is going to be equal to the term before it times a fixed amount so the second term is equal to the first term times 3 and we're summing them in a sequence you're just just looking at it it's an ordered list so to speak but here you are actually adding up the ordered list so what we just saw in this example is y a G of 1 what a geometric series is but also a famous example of how it's useful and this is just scratching the surface if you were to go further and finance or in business you'll actually see a geometric series popping up all over the place