Everyone, in previous videos, we saw how to use the general formula to figure out the terms of a sequence. For example, if I had something like , then the whole idea was I can grab these indexes, 1, 2, 3, 4, 5. I plug them into this equation here to get my outputs. What we saw is that the first five terms of the sequence were 2, 4, 6, 8, 10. What I'm going to show you in this video is that sometimes you may be asked to use or write a different kind of formula called a recursive formula. I'm going to show you the difference between general and recursive formulas, and, basically, what it is is that recursive formulas tell you how to find terms, your terms, but not based on that you plug into the equation, but it's actually based on the previous terms, the terms that go before in the sequence. I'm going to break down the difference, and we'll do some examples together. Let's get started.
So like general formulas, recursive formulas also tell you how to calculate terms of a sequence. We had something like . You plug these values in and you get your numbers that way. But now let's look at a different type of formula. This formula says , which is the next term in the sequence, is equal to . What this really what this is, is it's basically just the previous term in the sequence. It's the index subtracted by 1, so it's the previous term. So really, let's take a look here. Let's say you have this formula, , and you have the first term in the sequence, which is . Let's calculate the next few terms of the sequence using this formula.
Alright. So if I wanted to calculate , what this formula tells me is I'm going to have to use the previous term in the sequence, , and I'm just going to have to add 2 to it. Alright? And so what this is, it's just 2, so . Now let's calculate the next term, this . What this formula says is that to calculate , I need , and I have to add 2 to it. We actually just calculated what 2 is, that's just the 4 that we just calculated. So, I'm going to have to take 4, add 2 to it, and now I get 6. So, now I have 4 and 6. If I wanted to calculate , you'll see the pattern here: I have to know what is, and so on and so forth. So, you're just going to get 6 plus 2, and you're just going to get 8. If we continue on this pattern, what you're going to see is that we get 4, 6, 8, 10. Notice how we have actually ended up with the same exact numbers that we did when we did this with the general formula, but we just use a totally different formula to get there. So the basic difference between general and recursive formulas is that for the general formula, you're going to need to plug into this equation to get the nth term. So you need to know what is. And for a recursive formula, you just need the previous term in the sequence, this , to get what the next term is. That's the main difference.
Now, I want to point out that you might be thinking, well, isn't the general formula always going to be better? And not necessarily. Sometimes you may be asked to just find the next few terms of a sequence, and finding the general formula, if you're not given it, might be really hard. So it's just easier to sort of tell what the pattern is between these numbers. Hey. Look at these. All these are just increasing by 2, so I'm just going to continue on that pattern. So it's not that one is always better than the other. It just depends on what you're given to ask. Let's go ahead and take a look at some examples here and work this one out together.
Alright? So given this recursive formula and the first term of each sequence, I want to find the next three terms in the sequence. Alright? So I'm told here that in example , , that's the previous term, and then I'm going to have to add 3 to it. And I'm already told what is, that's just equal to 1. Let's use this formula to now find the 2nd term. The 2nd term says, or this formula says, in order to find , I'm going to have to take 2, multiply it by the previous number in the sequence, which is , and then I'm going to have to add 3. But notice that we've already calculated that is 1. This is going to be . If you work this out, what you're going to get is 5. So the second term of the sequence is 5. Now let's take a look at the third one. This says, in order to calculate the next term, I'm going to have to take 2 and multiply it by the previous term and add 3. We've already figured out what is. It's just 5 over here. So this is just going to be , and that gives me 13. Now for the 4th one, this is going to be . You've already figured that out. That's just 13. So this is going to be , and this is going to equal 29. Alright? So those are the first four terms in the sequence. Notice how, if I just look at these numbers and try to come up with a general formula, it might be really tricky to do this. So if you're just asked to figure out the next few terms of a sequence, usually, the recursive formula is going to be really, really good for this.
Alright? So here are your answers. Let's take a look now at the second one. In fact, actually, pause the video and see if you can try it on your own. So in the second formula here, what I've got is I've got . And I've got , which equals 1. What's the second term? Well, just tells me that it's just going to be the index itself, in which case I'm using . So this is going to be . So in other words, this is just going to be . So that is . So now let's calculate . This is where . Right? So what I'm going to do is this formula says take the index, which is 3 in this case, and now multiply it by the previous number, which is . So what is that? So this is going to be , and that gives me 6. That's the 3rd term in the sequence. So now for , hopefully, you see the pattern now. This is where . You're going to grab the index, which is 4, multiplied by the previous term, which is . This is going to be , and this is going to be 24. Alright? That's going to be 24 over here. Alright? And that's how you use these, these recursive formulas to figure out the next few terms of a sequence. Hopefully, that made sense. Thanks for watching, and let's get some practice.