Everyone, welcome back. So in previous videos, we saw how to write recursive formulas for arithmetic sequences. For example, in this specific sequence, we saw this was the recursive formula: You take the previous term and then add 5 to it. But in some problems, you may be asked to write a formula for the general or the nth term in a sequence, and you may have to use it to find something like the 101st term in a sequence, something with an incredibly high index. Doing this with a recursive formula would be a nightmare because you'd have to calculate first the first one hundred terms. So instead, what I'm going to do in this video is show you how to write a general formula for this arithmetic sequence. Remember, these equations allow you to calculate any terms without having to know what the previous terms are. So I want to go ahead and show you the difference between these two, and we'll do some examples together. Let's get started.
General formulas are always formulas that contain n, whereas recursive formulas always contain an-1, the previous term in a sequence. Well, for a general formula for arithmetic sequences, it'll give you the nth term, and it's based on a couple of things. It's based on the first term in the sequence and the common difference, d, that lowercase d we've been working with. So your textbooks are going to do some lengthy derivations for this. I'm actually just going to go ahead and give you what the formula looks like. To calculate the nth term in a sequence, you're going to have to know the first term, a1, that's what we just said, plus d times n-1. I want to go ahead and make sense of this equation using the sequence that we have over here, 2, 7, 12, 17, so on and so forth. Remember, these are the first four terms in the sequence, a1, 2, 3, 4, so on and so forth. So notice how the difference between a1 and a2 to get from a1 to a2, you just have to add one multiple of d, the common difference. Now to get from 2 to 12, you have to go and start at the first term, and then you have to add 2 times d, that multiple or that common difference. And to get to the 4th term, you have to know what the first term is, and then you have to add 3 times d. So notice how there's a pattern going on here. To calculate the nth term, you always start at the first term, and then you have to multiply d by n-1. That's what's going on here in this formula. So for this specific sequence over here, the general formula would just be the first term in the sequence, which in this case is 2, plus the common difference, which we know from this sequence over here that it's 5, and then we multiply it by n-1. This is the general formula for this sequence over here. We can use it to calculate any terms, 2, 3, 4, whatever. Let's go ahead and calculate the 4th term of the sequence using this formula because we know from the recursive formula that it should be 17. So we're just going to sort of double-check here. So a4 would be a1, which is 2, plus 5 times n - 1, which is going to be 4 - 1. Remember, this is just going to be where n=4, so we're going to have to plug in 4 - 1. So this is just going to be 2 + 5×3, in which case you're going to get, 2 + 15, which is 17. And that's exactly what we should get here, what we should expect. So using this general formula, we found that the 4th term is 17. Alright? That's really all that's going on here. This is always going to be the formula that you start off with when you're writing a general formula for arithmetic sequences.
Let's go ahead and take a look at another example and work this one out together. So for this sequence below, we're going to write a formula for the general or nth term. So this just means we're going to write a general formula and use it to find the 101st term. So we have the sequence over here. I've got 2, 5, 8, 11, and 14. So let's just go ahead and write out our general formula equation, an = a1 + d*n-1. Now it might seem kinda scary at first because it's lots of letters that are going on here. But, really, all you need to know is you need to know what a1 is and d, that common difference. These are the only numbers you're going to plug in for because, remember, n is going to be your index. You don't plug in for that unless you're actually finding a certain term. Alright? So what is a1? Well, that's pretty easy because you just look at the first term over here. So this is just going to be a1. That's going to be the 2. Alright? So in other words, your formula is just going to be 2 plus now what's the common difference in this? Well, you may just take a look at the numbers, 2, 5, 8, 11, 14. There's a pattern going on, which is that each one of these things increments by 3. And that's going to be that common difference. That's what d equals. It's positive 3. So this is going to be 2 + 3. This is your general formula for this arithmetic sequence. That's all there is to it. You don't plug anything in for n just yet because this just shows you how to calculate the general term an. All right? So now that we have this general formula, now we can use it just to find the 101st term. All we have to do is just plug in 101 in for n inside of this equation. So what this says here is that we're going to take the first term, which is 2, and we're going to add the common difference of 3 times 101 minus 1. In other words, this is going to be 2 + 3×100. And so, in other words, the 101st term is really just going to be 3 times 100 plus 2, which is 302. Very easy. So if we had to do this via recursive formula, it would have taken us forever to do this. And that's why these general formulas are super powerful. Anyway, thanks for watching, folks. That's it for general formulas. Let's go ahead and get some practice.