Writing geometric series in sigma notation

So we have a sum here of 2 plus 6 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, to 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 three. So let me write that we're multiplying by 3 to go to 6. To 18, what are we doing? Well, we're multiplying by 3. To go to 18 to 54, we're multiplying by 3.

So it looks like this in this is indeed a geometric series, and we have a common ratio of 3. 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 0.

And so we have our first term, which is 2, so it's 2 times our common ratio to the kth power. So times our common ratio, 3 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 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, it'll 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 0. Let me just... 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 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 going to go up to k is... k is equal to 3. So that's one way that we could write this. There are 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 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 1, it's 1 minus 1; you get the 0th power, and so we're just... we're 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, it's 3 to the 4 minus 1 power, so it's still 3 to the third 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.