>> In this section and the next we are going to talk about two rather special sequences. Here's an example of the first one, and notice that the prevailing characteristic here in generating one term after another is that all we do is add three. One plus three is four, four plus three is seven, seven plus three is ten. So what that implies then is that the difference between succeeding terms is three for all of the terms. That's the prevailing characteristic for arithmetic sequences. That's what we are talking about here. Now notice this pattern. Ace of one. The first item in the sequence is one. To generate the second item in the sequence, one way to think about it is to take the first item and add that common difference. You see? And then for the third item we can take the second item and add another common difference. And for the fourth item we can take the third item and add another common difference. Now look at the number of common differences for each of these items. For the second item we have one common difference. For the third item we have two common differences. For the fourth item we have three common differences. Generally, or a little more generally, ace of one is the first item in the sequence is ace of one, plus no common differences. And then ace of two is ace of one plus one common difference. Ace of three has two common differences. Ace of four has three common differences. And then you can generalize it even further to the Nth term. You can say the Nth term would have, well, let's see, how many common differences? Well, you notice that this number, the number of common differences is one less than the number here. What's one less than N? Its N minus one. So it's ace of one plus N minus one times D, common difference. Now using that kind of model we can a lot of times generate the expression for the Nth term of an arithmetic sequence. And for this arithmetic sequence it would work like this. We understand that with arithmetic sequences we have this relationship. That's the Nth term is just the first term plus N minus one differences. And now let's plug in the information that we know for this sequence. Ace of one is one. And we have N minus one differences, but the common difference we know to be three. And now just evaluate this expression and we have one plus, let's see, three times N, three N. Three times negative one, negative three. And collecting here we find ace of N to be three N minus two. Now let's see if that's not true. I have  let me just test it on one of these, one of these items in the sequence. We have, let's see, this is the first item, second, third, fourth, fifth. The fifth item is thirteen. Let's see if replacing N with five will give us that item. That is, is it true that ace subscript five is thirteen? Well, we would have three times five minus two, that's fifteen minus two thirteens. Yeah, this works out well. The general model can be used in problems like this. Any time we are identifying an arithmetic sequence we just bring that general model to bear on the problem. And they can have a lot of different looks. This one is pretty straight forward though, it says find a formula for the Nth term of the arithmetic sequence with a common difference of two and a first term of five. So we are given all of the important information. So generating the expression for the Nth term works like this. You say, well, the general model for arithmetic sequence is that the Nth term is the first term plus N minus one differences. Then we fill in the information we know, we know that the first term is given as five, and the common difference is given as two. Then we just evaluate, we just simplify the expression. We have five plus two N minus two. And collecting constants on the right the expression is two N plus three. One important aspect of this expression, when we can see a distinct coefficient of the counting mechanism, it turns out that that coefficient of the counting mechanism is the common difference involved in the sequence. Now we are told that the difference is two here and we have a two here. But if we are not told if we have an expression then that coefficient of the counting mechanism will be the common difference. And it kind of makes sense because every time we count here, every time we add another value to N, then we are multiplying that value by two. You see, two times one, two times two, two times four, two times five, and for successive terms generated by this expression all we are doing is adding another value of two every time we pop in other value for N.