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so we have a sum here of two plus six plus 18 plus 54 and we could obviously just evaluate it add up these numbers but what I want to do is I want to use it as practice for rewriting a series like this using Sigma notation so let's just think about what's happening here let's see if we can see any pattern from one term to the next let's see they go from two to six we could say we're adding four but then we go from six to eighteen we're not adding four now we are now adding twelve so it's not an arithmetic series let's see what of maybe it's a geometric so to go from two to six what are we doing well we're multiplying by we're multiplying by three so write that we're multiplying by three to go to 6 to 18 what are we doing where we're multiplying by three to go to 18 to 54 we're multiplying by three so it looks like this in this is indeed a geometric series and we have a common ratio of three so let's rewrite this using Sigma notation so this is going to be the sum and we could start oh there's a bunch of ways that we could write it we could write it as let's start with K equaling zero and so we have our first term which is 2 so it's 2 times our common ratio to the K power so times our common ratio 3 to the K power so before I even write how many terms we have you're how high we go with our K let's see if this makes sense when K is equal to 0 there's going to be 2 times 3 to the 0th power so that's 2 times 1 so that's this first term right there when K is equal to 1 will be 2 times 3 to the first power well that's going to be 6 and then when K is so this is K equals the 0 let me do this in a different color so this is K equals 0 I say different color and then I do the same color all right so this is K equals 0 this is K equals 1 this is K equals 2 and then this would be K equals 3 which would be 2 times 3 to the third power so 2 times 27 is indeed equal to 54 so we're going to go up to K is K is equal to 3 so that's one way that we could write this there's other ways that you could write this you could write it as so we're going to still do we have our first term right over here but for example we could write it as our common ratio and I'll use a different index now let's say to the N minus 1 power and instead of starting at 0 I could start at N equals 1 but notice it has the same effect when you say N equals 1 it's 1 minus 1 you get the 0th power and so we're just increasing all of the indexes by 1 so instead of going from 0 to 3 we're going from 1 to 4 and you could verify that this is still going to work out because when n is equal to 4 is 3 to the 4 minus 1 power so that's still 3 to the 3rd power which is 27 times 2 which is still 54 so this is N equals 1 that is N equals 2 that is N equals 3 and that is N equals 4 but either way these are ways that you could write it using Sigma notation