Let's now look at the sum of a geometric series. Say we have 1 plus 2 plus 4 plus 8 and so on, all the way to 1,024 to add up. The sequence we are adding in the series it is a geometric progression. It begins in one and the common ratio is two. It is a doubling sequence. The general term is bn which is two to the power of n minus one, and we want to add all the terms all the way to 1,024. We need to work out how many terms that is to start with. So, a 1,024 is two to the power of 10. We are adding two to the zero, two to the one, all the way to the power of 10. That's 11 terms. So, we want to add bn from n being one all the way up to n being 11, so it's 11 terms. So, let's say for now, we're going to call this sum big S. I'm going to write it on top as well. So, that is our sum. It's quite a clever little trick to work out this sum. We're going to multiply the sum itself by the common ratio, by two, because this is a doubling geometric progression. So, we're going to multiply that sum by two, and that's going to help us, let's see. So, two S. For us to appreciate the pattern, let's use the sum at the very top. Twice the sum is going to be twice every term. So, I'm going to add two, which is twice of one. I'm going to add four, which is twice of two, twice of four, twice of eight, all the way to twice of 1,024. Okay. Now, let's compare the two, let's compare the sum and twice the sum. Most numbers are the same. As in I've got numbers from one to four, all the way to 1,024, and two S goes from two, to four, and so on. Here, there's a 1,024 as well, which is the term that comes before the 2,048. I'm going to rewrite it there. Now, the clever bit is to subtract the two sums. So, two S goes from 2 plus 4 plus 8 all the way to 2,048. S is 1 plus 2 plus 4 and so on all the way to 1,024. Now, if I subtract S from 2S, I'm going to subtract 1,024 from that 1,024. I'm going to subtract all these numbers here in the middle and these two, and these two, and these two. So, those are all that got canceled out and I'm going to end up with 2,048 minus the 1. So, 2S minus the S is going to end up being 2,048 take away 1. But all we did was to start with twice the sum and subtract the sum. That means, I just got one sum here. So, this is the value of the sum and that is going to be 2,047 from here. It's done. We didn't have to add up all those numbers. This is the shortcut. The good news is that this also applies to any geometric progression not just to the one we saw and the same method is done. I'm going to write down the general case. So, if we are doing the sum of a geometric progression that starts in one and then this ratio r. These are going to be the terms. I'm going to add n terms of this series. So, the last term is r to the n minus one. The trick I did was to multiply the sum by the common ratio. So, I'm going to do r times S. If I do that, I have the one on the sum times r. So, I'm going to have r. The r of the sum times r is going to be r squared, and so on. Every term above is going to be multiplied by r. But I have as many terms in one of the other. So, the last one is r to the power of n. I'm going to subtract, I'm going to do rS take away S. That means, these here will cancel out, they will subtract to zero. The same with everything in the middle. I'm left with r to the n take away the one. Now, let's look at the left-hand side. This is r times S minus S. So, that is r takeaway 1 multiplied by the sum. On the right hand side I've got r to the n minus 1. I can divide both sides of the equation by r minus one. So, I have the S is r to the n minus 1 divide by r minus 1. Of course, if r isn't one. If r was one, we didn't need this formula to add it up. It wouldn't be an interesting geometric progression. So, this is the sum of n terms of a geometric progression. I'm going to write S with a little n underneath. Of course we are using the series notation. So, the series notation we have the big sigma. So, I'm going to rewrite that as the sum of the powers of r, all the way from zero to n minus one. Remember, we were doing one plus r plus all the way to r to the n minus one, so those are the limits of my sum. This is given by the formula we just wrote which is, r to the power of n takeaway one over r takeaway one, provided r isn't one. So, that is the sum of a geometric series. Let's just see that formula applied to the sequence we just added up. When r was two, the sum of two to the k, k running from 0 to 10. I know we wrote k going from 1 to 11 on the other example, but we're using the expression of two to the k here so this is the same value. This is using the formula two to the n, n here is 11 minus one, two minus one, and that is two to the 11 takeaway our one because of the nominators is one. That is 2,048 takeaway one as we had calculated before. Okay? Let's look at another couple of examples, say we've got the sequence 0.3, 0.06, 0.012, and so on. It is a geometric progression, the common ratio is 0.2, and you would verify that by doing 0.06 divided by 0.3, and that is 0.2, and it's the same as 0.012 divide by 0.06. Okay? So, it is a geometric progression with ratio 0.2. Let's add the first five terms of that series. So, we want to add an when n goes from one to five. Here, what we do is, we note that this series is not exactly powers of the ratio, but the ratio is 0.2. The general term of the sequence, we're are going to write it up here. The general term for the sequence is an equals 0.3 times 0.2 to some power. Well, when n is one, I'm not multiplying by 0.2, is only when n is two. So, I'm going to do n minus one. So, the sum I've got here, the series I've got here is 0.3 times the sum of 0.2 to the n minus one. So, that is 0.3 times 0.2 to the power of five take away one over 0.2 take away one, which equals to 0.37488. So, that is the sum of the first five terms of that geometric progression. One more example. Let's do 0.1 minus 0.05 plus 0.025, and so on. This is the series we want to add up. Now, I want to add up the first five terms of that geometric progression. Let's check that it is a geometric progression. So, the terms are 0.1 minus 0.05, 0.025. So, let's do the ratios, minus 0.05, this is minus a half and that is the same as 0.025 divide by minus 0.05. Okay? So, we want r to be minus one half. So, the sum we're calculating is 0.1 times the sum of powers of minus a half. We're running this from zero to four, that's five times. Now, this is going to be 0.1 times the formula for the sum, which is minus one half to the power of five, is five terms, take away one divide by minus one half takeaway one. That is 0.06875.