 So in previous videos, we had talked about the idea of a sequence where we're given a list of numbers and we can describe that sequence using some general form or maybe a recursive relation, what have you, but oftentimes the genesis of our discussion sequence is with the sequence itself. We see a list of numbers, but in order to do any type of analysis or calculus with that sequence, we often need a general form or maybe a recursive relationship. And so let's try to analyze a couple of sequences and see if we can find a pattern to them. So you look at this first sequence right here. E, E squared over 2, E cubed over 3, E to the 4 over 4, E to the 5 over 5. And so some patterns I notice here is that in each of these terms, there appears to be some E, the number E of course, that's going on here. It looks like it's in the numerator, right? And the power seems to be increasing by one each time, right? We see that as a recursive relationship. We see E to the first, E squared, E cubed, E fourth, E to the fifth. And as such, we then think that our general form, A sub n, is going to have some power of E to it, E to the end. But we also recognize that each of these is a fraction. The first one can be written as E over 1. In which case if you look at the denominators, you always see a 1, 2, 3, 4, 5. The denominator seems just to be increasing by one each time and it seems to match the exponent of E. In which case we get this general form, A n equals E to the n over n. That one's not so bad. Let's look at example B right here. So we have a sequence again, kind of like the first one, we can regret this as fractions 1 over 1, 1 over 3, 1 over 9, 1 over 27, 1 over 81. The numerator always seems to be a 1, so I can use that aspect here, B sub n. B sub n will equal 1 over 1 over what? Well, if we look at the denominators, what's the connection here? 1, 3, 9, 27, 81. These appear to be powers of 3 to the 0, 3 to the 1st, 3 squared for 9, 3 cubed, 3 to the 4th. And so we're going to get powers of 3, 3 to the n, but we have to be careful there, right? The first term, which is right here, actually has the 0 with power. And then the second term has the first power, the third term has the second power, the position we are in the sequence actually seems like we're actually taking the previous power of 3. And so really B to the n should be 1 over 3 to the n minus 1. Our power of E is actually 1 less than we are in the sequence. It's reasons like this where it actually might be advantageous to start your term, your sequence with the 0th term. One could take that, but that's a convention we're not going to follow for right now. Looking at the third sequence, C right here, we have the sequence 1, 3, 5, 7, 9. This seems to be the sequence of odd integers, and thus you're increasing by 2 each time. So if you think of your sequence, Cn, well C1 appears to be 1. And then the next term, Cn, seems just to be the previous term, Cn minus 1 plus 2. I mean, that's how we get this odd sequence of numbers. And so by adding multiples of 2 each time, we can actually see the following pattern. Cn is going to start off with a 1, and then we add a multiple of 2 to it each time. 2, but the multiple is 1 less than we are right now. And so we see that we could do, if you look at this sequence real quick just to check it, the first term you won't add any multiples of 2. The second term you'll add a multiple of 2, so you get 1 plus 2 which is 3. With the third term, when it equals 3, you're going to add 1 plus 2 times 2, right? In which case you get 4 plus 1 which is a 5. So we can see that this thing matches up here, 1 plus 2 times N minus 1. I'm kind of using the fact that this isn't actually what's called an arithmetic sequence. It increases by a constant amount each time you go throughout the sequence there. Now we can simplify this formula of course by distributing the 2. You're going to get 1 plus 2N minus 2 which simplifies to be 2N minus 1. It's a nice little formula for the sequence of odd integers. We're actually going to use this one a lot. So this is one we want to come back to in the future. I'm looking at 1, 4, 9, 16, 25. That looks like the sequence of squares. So we could say dN is equal to N squared. That's a nice one. And then the last one, e here, we have 1 negative 1 half, 1 third, negative 1 fourth, 1 fifth. So this kind of seems like something we saw before with the e's, right? e sub N, we have these fractions. It's always 1 over where we are in the sequence, 1 over N. But there is this issue about the sign, right? We have a positive one, then a negative one. Then we have a positive one again, then a negative one, then a positive one. It switches back and forth between positive, negative, positive, negative, positive, negative. This behavior we're seeing right here is often referred to as an alternating sequence because of the alternation of signs. It switches positive, negative, positive, negative. And so this is something difficult to have for a continuous function because there is no next term when you look in an interval of real numbers. Like if you take the number pi, what's the next number? You can find a bigger number, but it's not the next one. We can find a bigger one, you can find one that's still closer to pi. But with sequences, there is always a next term, a next term, a next term. And so we can describe things like alternating sequences in a way that don't really make sense for continuous functions. And so what's going to happen here is we are going to take powers of negative 1. Notice, whoops, if we take negative 1 raised to the N plus 1 power, I'm going to rewrite this thing right here. So if we take negative 1 to the N plus 1 power times that by 1 over N, notice what happens when N equals 1, then the power of negative 1 you're going to get is a 2, which is a positive 1, particularly that's positive. When you take N equals 2, you're going to get negative 1 to the third power. That's an on power of negative 1, so you get a negative 1. And that matches up with that sequence we saw, positive, negative. The next one, if you do N equals 2, sorry, N equals 3, you're going to take negative 1 to the fourth power, which is a positive 1. And so then the pattern continues, positive, negative, positive, negative, positive, negative, positive, negative. And so this idea of a negative 1 to some power of N is how we can capture alternating sequences. If you start positive then negative, you're going to take N plus 1 as your power of negative 1. If you start negative then positive, you actually can just get away with negative 1 to the N, like so. And so this gives us some examples of how we can actually build equations, formulas for upon the pattern that we see here. These patterns can be difficult to find sometimes, but be patient with yourself and we can often derive a formula for these sequences that we run across.