 Since a series is an ordered sum, we can try to find the sum of the series. For example, suppose I want to find the sum of the series, i goes from 3 to 7 of 3i plus 4. So the series is the sum of the terms calculated using the formula 3i plus 4 for i values from 3 to 7. So we can find what those terms are. We can then add these terms together, which gives us the sum of the series. And so this leads to some problem. Given a series, what is the sum? Now if the series is finite, we can just add the terms together. But maybe trying to add all these terms together is going to be a little tedious, so maybe there's an easier way. So let's consider that series. The sum from i equals 1 to 100 of i is going to be the sum. Now because this is a sum, we can order the terms any way we want to. Why does reversing the order help us? Well if you notice, if I sum vertically, the sum of these first two terms is 101. And if I sum vertically, the sum of these next two terms is 101. And if I sum these two terms vertically, their sum is 101. And you should convince yourself that the sum of these other vertical pairs of terms will also be 101. Now that means that twice the sum of the series is the sum of a whole lot of 101s. How many 101s? Well here's one way that we might approach this question. This 101 corresponds to the entry 1 in our series. It's the first 101. This 101 corresponds to the entry 2, and it's the second 101. This entry corresponds to 3, and it's the third 101. And in general, every 101 corresponds to an entry up here that's also the same as its term number. So that means this last 101 is actually the 100th 101 in this sum. And so this sum is 101s added together. And that tells me the sum of the series itself is going to be 1 half 100 times 101 or 5050. Well let's try a different sum. So it's helpful to write down the first few and last few terms of the series. Since the terms of the series are 2i plus 5, and i goes from 10 to 35, the first few terms are going to correspond to i equals 10, 11, and 12. Meanwhile the last term corresponds to i equals 35, so it's going to be, and the term before it is going to correspond to i equals 34, so it will be, and the term before that corresponds to i equals 33, so it's going to be. Now if I reverse the sum, then add down the columns, we see that every pair of terms adds to 100. So twice the sum of the series is the sum of a whole lot of hundreds. Now for the hard part, how many terms are there in this sum? Well we might notice the following. The first one term corresponds to i equals 10. The second two term corresponds to i equals 11. The third three term corresponds to i equals 12. And the last term corresponds to i equals 35. And so now we ask ourselves, what's the relationship between the term number and the value of i? And after staring at this for a while, it seems that the term number is 9 less than the value of i. And so this last term must be the 26th term. And so what we have is we have 100 added together 26 times. And so twice our series sum is 26 times 100. And so our series sum itself will be half this amount. And we can generalize this process to produce a general formula for the sum of an arithmetic series. And for this formula, I say what we say for all formulas, it's probably not worth memorizing the formula itself. It's more important to memorize the process. We reversed our terms, added them together, found the number of terms, and by adding everything together found twice the actual sum, so we divided by 2 to get the sum itself. If you do insist on memorizing a formula, here's a way to think about it. We added the first and last terms together, then we multiplied by the number of terms we added, which gave us twice the sum of the series, so we divided this product by 2. How about trying to sum a geometric series? Suppose we have the geometric series, the sum from 1 to 100 of 1 tenth to power i. So let's write down the first couple of terms of the series and the last couple of terms. And here the thing to notice is that if we multiply the entire series by the common ratio 1 tenth, what that does is multiply every term by 1 tenth, and so we get... and many of the terms in the two expressions are the same. And so this means that if we subtract the one from the other, most of the terms will cancel out. So let's do that. Let's find the difference between the two expressions. What makes this work is that most of the terms in the first series will be canceled by most of the terms in the second series. So when we subtract this term 1 over 10 to the second, we'll cancel out with this term 1 over 10 to the second. The 1 over 10 to the third in the subtracted series will cancel out the 1 over 10 to the third in the original series, which is the term after this 1 over 10 to the second, and so on until this 1 over 10 to the 100 in the subtracted series cancels out this 1 over 10 to the 100 in the original series. And that means that when we find the difference, the only terms that are left are the 1 over 10 in the original series minus the 1 over 101 in the subtracted series. And though I really detest bumper sticker mathematics, there is one phrase that's useful. If in doubt, factor out. We see that both of the terms on the left-hand side have this factor of the sum from 1 to 100 of 1 tenth to the i, so let's remove that as a common factor leaving us with, and that allows us to solve for the sum of the series. So we could try this on a different series. So let's take our series, and again we'll write out the first couple of terms as well as the last term. If we multiply everything by 1 third, that's going to get us a copy or a near copy of the series. If I subtract everything but the first term of the first series and the last term of the second series is going to cancel out, and so I'll have 2 thirds the sum equaling a difference, and so I know the sum of the series itself. And as with the sum of the arithmetic series, we get a formula that is more or less useless. Again, it's more useful to focus on the process that gave us this formula. We took our series, multiplied it by the common ratio, which essentially shifted all of our terms one space over. We then subtracted, which eliminated all but the first and the post-ultimate term, the term after the last one we actually wanted to sum. This then gave us a nice simple equation that we could solve for the sum of the series. Again, if you insist on spending your time memorizing a formula, then one way we can look at this is that the sum of the geometric series is the difference between the first term and the post-ultimate term, divided by 1 minus the common ratio.