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Current time:0:00Total duration:2:48

Worked example: finite geometric series (sigma notation)


Video transcript

let's take let's do some examples we're finding the we're finding sums of finite geometric series and let's just remind ourselves in a previous video we drive the formula where the sum of the first n terms is equal to our first term times 1 minus our common ratio to the nth power all over 1 minus our common ratio so let's apply that to this finite geometric series right over here so what is our first term and what is our common ratio well and what is our n well some of you might just be able to pick it out by inspecting this year but for the sake of this example let's expand this out a little bit this is going to be equal to 2 times 3 to the 0 which is just 2 plus 2 times 3 to the first power plus 2 times 3 to the second power I could write first power there plus 2 times 3 to the third power and we're going to go all the way to 2 times 3 to the 99th power so what is our first term what is our a well a is going to be 2 and we see that in all of these terms here so a is going to be 2 what is our well each successive term as K increases by 1 we're multiplying by 3 again so 3 is our common ratio so that right over there that is our let me make sure that we that is a and now what is n going to be well you might be tempted to say well we're going up to K equals 99 maybe n is 99 but we have to realize that we're starting at K equals 0 so there is actually 100 terms here notice when K equals 0 that's our first term when K equals 1 that's our second term when K equals 2 that's our third term when K equals 3 that's our fourth term when K equals 99 this is our hundredth term 100th term so what we really want to find is S sub 100 so let's write that down s sub 100 for this geometric series is going to be equal to 2 times 2 times 1 minus 3 to the hundredth power to the the hundredth power all of that all of that over all of that over 1 minus 3 and we could simplify this I mean at this point it is arithmetic that you'd be dealing with but down here you would have a negative 2 and so you'd have 2 minus 2 R 2 divided by negative 2 so that is just a negative and so negative of 1 minus 3 to the hundredth that's the same thing this is equal to 3 to the hundredth 3 to the hundredth power minus 1 and we're done