 Sequences in series form an important topic in higher mathematics. A sequence is an ordered set of real numbers where the kth term of the sequence is designated a sub k. This set could be finite or infinite, so there are finite sequences, but they're not very interesting. Since sequences are ordered, changing the order changes the sequence, so the sequence 1, 2, 3, 5, 8, 13, and the sequence 13, 8, 5, 3, 2, 1, even though they consist of the same numbers, because those numbers are in a different order, these two sequences should be regarded as different sequences. It's possible for the terms of a sequence to be defined by some formula. We usually indicate that by the notation a sub n equals f of n, where f is the formula and n is the index, it identifies which term of the sequence you're looking at. For example, suppose I want to write the first five terms of the sequence an equals f of n, where f of n is equal to n squared minus 1. Since an equals f of n, then a1 is my function evaluated at 1, which is 0. a2 is my function evaluated at 2. For a3, I'll evaluate my function at 3, 4, and 5. So this gives me the first five terms of the sequence. Since there are additional terms we haven't calculated, we should use the ellipsis, the dot, dot, dot, to indicate this. Now there's many, many, many, many, many, many, many different sequences, but there are a couple of important types of sequences. So let's consider what the sequence zoo looks like. Since a sequence is an ordered set of elements, you should be able to find the n plus first term of the sequence by knowing something about the preceding n terms of the sequence. In the worst case scenario, you have to know all of the preceding terms of the sequence in order to figure out the next term of the sequence. And that makes for a very difficult problem. But we can take a look at a simpler problem. There are two important possibilities. First, the n plus first term is a constant difference more than the preceding term. And second, the n plus first term is a constant multiple of the preceding term. In the first case, where the n plus first term is a constant difference more than the preceding term will end up with an arithmetic sequence which we'll define as follows. If the difference between any two consecutive terms of a sequence is a constant, then the sequence is an arithmetic sequence and the difference is called the common difference. And it turns out that we can show the following. Suppose I have an arithmetic sequence where you'll notice that we're actually starting our sequence with a zero. Let this be an arithmetic sequence with a common difference of d, then a n is a zero plus n d. You'll actually prove this in one of the problems. So let's see if we can find the formula for the nth term of the sequence that starts 1, 3, 5, 7, assuming it is arithmetic. So if the sequence is arithmetic, then we need to know the initial term and the common difference. The initial term is easy. As for the common difference, because it's the same for all terms of the sequence, we could just find the difference between the first two terms. But, and I know this is hard to believe, not everything on the Internet is true. So rather than assuming this is arithmetic because the problem says it's arithmetic, let's find the difference between consecutive terms and verify the claim. And we find that the difference between consecutive terms is always 2, which supports the claim that the sequence is arithmetic. At this point we can write the formula for an arithmetic sequence. It's going to be a zero, our initial term, plus n times the common difference. How about this second type of sequence where the n plus first term is a constant multiple of the preceding term? This gives us what's called a geometric sequence, and we'll define it as follows. If the ratio between two consecutive terms in a series is a constant, then the sequence is a geometric sequence and the ratio is called the common ratio. And as with arithmetic sequences, there's an easy formula for the nth term of a geometric sequence. Us have a geometric sequence with common ratio r, then an is a zero times r to power n. And again, this is a theorem that you'll be able to prove in one of the exercises. So for example, let's say I have a geometric sequence, and the thing that defines a geometric sequence is its initial term and the common ratio. Well, the initial term is just the first term of the sequence, 24. As for the common ratio, if this sequence is in fact geometric, then the ratio between consecutive terms of the sequence should be the same. So let's find that ratio. And we find that the ratio between every term and the term preceding it is one half, which supports the claim that the sequence is geometric. The common ratio is one half, and so this allows us to write the formula for the sequence. There's one more important type of sequence. If the terms of a sequence are alternately positive and negative, the sequence is alternating. For example, let's consider three sequences, sine of 2n, minus 2 to power n, and minus 1 to power 2n, and let's see which if any of these are alternating. We can go ahead and try to find the first few terms of each sequence. For an equals sine of 2n, we have for n equals 0, 1, 2, 3, and 4, and we see that the signs are sometimes positive, sometimes negative, but they do not alternate, and since the signs do not alternate, this is not an alternating sequence. For bn equals 2 to power n, we can find the terms of the sequence for n equals 0, 1, 2, and 3, and we find. And these do appear to alternate between plus and minus, but let's try and defend our conclusion here. So the thing we might notice is that if n is even, minus 2 to power n will be the product of an even number of negative factors, which is going to be a positive number. On the other hand, if n is odd, minus 2 to power n will be the product of an odd number of negative factors, which will be negative. So we might say the following, since the terms of the sequence bn equals minus 2 to power n will be positive if n is even, and negative if n is odd, then the sequence is alternating. And if we take our last sequence and evaluate it for n equals 0, 1, and 2, we find that the terms are all equal to 1, which does not appear to be alternating. And again, let's see if there's a reason for that. And here we might do a little bit of algebra, since cn equals negative 1 to power 2n is really negative 1 squared to power n, which is 1 to the n. These terms of the sequence will be always positive, and so the sequence is not alternating.