Hey, everyone. Welcome back. So throughout our discussion on sequences, we've often run across ones that look like this with numbers 2, 7, 12, 17, where the numbers increase by the same amount each time. In this case, each one of these numbers increases by 5. And we've also seen the ideas behind recursive formulas, which are formulas that tell you what the next term is based on what the previous term was. But we're going to put these ideas together in this video because it turns out that these types of sequences that we've already seen before have a special name. These are called arithmetic sequences, and I'm going to show you how to write recursive formulas for them. Turns out it is a very straightforward formula that we'll see. It only just depends really on one variable. So I want to break it down for you, and we'll do some examples together. Let's get started.

So an arithmetic sequence is really just a special type of sequence where the difference between terms is going to be the same number. So notice how in this sequence over here, 2, 7, 12, 17, the first term in the sequence is 2, and then each term after that increases by 5. The difference between any number and the previous number is always 5. Now this difference, we call the common difference, and the letter we use for this is lowercase d. So in this sequence over here, d equals 5. And we can use this common difference to set up a recursive formula to find the next few terms in the sequence. And, basically, all it is here is you're just going to set up a recursive formula, which remember says that the next term is the previous term, and then we're going to do something to it. So in order to get the next term in the sequence over here, notice the pattern. I just take the previous number, and I just have to add 5 to it. So a recursive formula for this sequence is just that a_{n} equals a_{n-1} plus 5. That's really all there is to it. This is the recursive formula that tells us the next few terms in this sequence. Alright? Now all we have to do is just specify what the first term in the sequence is, which in this case is 2, but that's really how you set up a recursive formula.

The general sort of format for this is that the new term is equal to the previous term plus d, whatever that common difference is. In this case, it was equal to 5. That's really all there is to it. All of your arithmetic sequences will always have recursive formulas that look like this. Alright?

Let's go ahead and take a look at our first example. In this first example, we're actually given what the recursive formulas are. So over here, in example a and b, what we want to do is we want to write the first four terms. Alright. So let's get started. So we actually already have what the first term is, which is a_{1}, which is equal to 3. And I want to calculate a_{2}. So a_{2} says that I'm going to take the previous term in the sequence, a_{1}, and I'm going to have to add 4 to it. And if I do that, that's just going to be 7. Right? Because I have 3 plus 4, and this is going to equal 7. So now, a_{3} is going to be if I take the previous term a_{2} and then add 4 to it. So, in this case, I just calculated what that is. That's just 7. So that's going to be 7+4, and this is going to give me 11. And I can do the same thing for a_{4}. This is going to be a_{3} plus 4, which in this case is just going to be 11. So this is going to be 11, and then plus 4, which will give me 15. So it turns out that these are the first four terms in the sequence. We've got 3, 7, 11, and 15. Alright? So I just have to take the previous term and add 4 onto it. Alright. So let's take a look now at the second example. Here we have a recursive formula that says that the new term is the previous term, but now we're going to subtract 6. And we're going to start off with the first term being 9. So let's take a look here. So a_{1} is equal to 9. If I wanted to calculate the second term, what this formula says is I'm going to have to take the first term and then subtract 6. But I know what the first term is. It's 9, so 9 minus 6 is equal to 3. Let's take a look at the third term. a_{3} says that this is going to be a_{2} minus 6. I already know what a_{2} is. It's equal to 3, so 3 minus 6 is negative 3. And a_{4} is just going to be a_{3}, which I just calculated was negative 3 minus 6. So this is going to be negative 3 minus 6, and this gives me negative 9. So notice how each one of these numbers is continuously decreasing by 6 each time. So sometimes your common difference can be a positive number, like, plus 4, and sometimes it could be a negative number, like, minus 6. That's perfectly fine here. Right? So here, d equals negative 6. Here, d equals 4, and you can still calculate the next few terms in the sequence. Alright? So in these problems here, we were already given what the recursive formula was and asked to find the terms, but sometimes you actually might be given the opposite. Sometimes you may have to actually write a recursive formula from a given sequence of terms. And whenever you are trying to write a recursive formula, we're just first going to have to find that common difference, that lowercase d. Let me show you how to do this step by step here. So in this case, we're going to write a recursive formula for the sequence 2, 5, 8, 11, 14. Notice how this is definitely an arithmetic sequence because the difference between each term and the next one is always the same number. Notice how we're just adding 3 to each previous term to get the new one. So what you're going to do here is to write a recursive formula. Remember that it's going to look something like this. Right? So your a_{n} is just going to be a_{n-1} plus the common difference. So the first step is actually to find that common difference by subtracting any 2 consecutive terms. And, really, it doesn't matter if you do 5 minus 2 or 8 minus 5 or 11 minus 8. It's gonna always be the same number. Now always make sure that you're that you're subtracting the next term minus the previous term because if not, you're going to get sort of a negative number. Right? So in this case, what you're going to do here is every term is adding 3, so that means d is equal to 3. In other words, 5 minus 2 is equal to positive 3. Right? So that's d. It's 3. So now we just plug it basically into this formula over here, which says so that the next term is going to be the previous term plus 3. Now are we done here? Well, the problem with this formula by itself is that if I just told you, hey. Take the previous term and add 3 to it. But I didn't tell you where to start, You wouldn't know what the next terms are. So whenever you're writing the formulas for recursive formulas, you always want to write the recursive formula, including the first term. You always have to specify what the first term that starts off the sequence because otherwise, you don't know where to start. Right? So this is a_{n} equals a_{n-1} plus 3, and the first term that you start off with is going to be the number 2. Using these two pieces of information, I could recreate any term in the sequence over here. Alright? So that's how to deal with recursive formulas and arithmetic sequences. Thanks for watching. Let's get some practice.