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## Advanced sigma notation

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# Finite geometric series in sigma notation

## Video transcript

- [Voiceover] So we have
sum here of two plus six plus 18 plus 54. And we can obviously just
evaluate it, add up these numbers. But what I want to do is
I wanna 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, to go from two to six, we could say we are adding four, but then when we go from six to 18, we're not adding four
now, we are now adding 12, so it's not an arithmetic series. Let's see, maybe it's a geometric. So to go from two to
six, what are we doing? Well, we're multiplying by three. So, let me write that,
we're multiplying by three. To go to six to 18, what are we doing? Well, we're multiplying by three. To go to 18 to 54, we're
multiplying by three. So it looks like 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, well, 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 two, so it's two times our common
ratio to the kth power. So times our common ratio
three to the k power. So before I even write how
many terms we have here or how high we go with our k,
let's see if this makes sense. When k is equal to zero, this
is gonna be two times three to the zeroth power. So that's two times one, so that's this first term right there. When k is equal to one,
that'll be two times three to the first power. Well, that's gonna be six. And then when k is, so
this is k equals zero. Let me do this in a different color. So this is k equals zero,
I say different color, and then I do the same color. All right, so this is k equals
zero, this is k equals one, this is k equals two, and then
this would be k equals three, which would be two times
three to the third power. So two times 27 is indeed equal to 54. So we're gonna go up
to k is equal to three. 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 gonna 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 one power. And instead of starting at zero, I could start at n equals one, but notice it has the same effect. When you say n equals
one, it's one minus one, you get the zeroth power. And so we're increasing
all of the indexes by one, so instead of going from zero to three, we're going from one to four. And you could verify that this
is still going to work out, 'cause when n is equal to four, it's three to the four minus one power, so it's still three to the third power, which is 27 times two which still 54. So this is n equals one,
that is n equals two, that is n equals three,
and that is n equals four. But either way, these are
ways that you could write it using sigma notation.