everyone. Welcome back. So in this example, we're going to do something very similar where you have the first five numbers of a sequence, negative 2, 4, negative 6, 8, negative 10, and so on and so forth. And in this problem, we're asked to again write the general formula for the nth term, and we're going to use this to calculate the 18th term. So, again, these formulas are really useful for calculating really, really high terms because rather than having to sort of follow-up the pattern for 18 terms, we're going to be able to figure out a formula for \( a_n \). Right? That's just going to be some formula over here. And then we're going to use this to figure out the 18th term by basically just plugging in 18 for \( n \). Alright? So let's go ahead and get started here. Remember, there are lots of things to consider when writing general formulas. Let's look at the pattern of numbers. So do we see fractions? We don't see any fractions. Do we see numbers that increase by the same number each time? Well, if you take a look here, what happens is that from negative 2 to 4, that's an increase of 2. I'm just looking at the number, not the sign. From 4 to 6, I also have an increase of 2, 6 to 8, increase of 2, and increase of 2. So I've got the same number that sort of it increases each time, but now I've got this other thing that's happening as well, which is that the signs alternate. I've got negative and then so I've got negative here then positive, negative here then positive, negative, and then so on. So, actually, I'm going to sort of combine some of the rules that I've seen before. So remember, whenever you have alternating signs, that always means that there's going to be a negative one to the nth power. And remember if the first term is is is, negative, that means that we're going to have an \( n \) and not \( n+1 \). So if the first term is negative, then it's just going to be negative one to the nth power. Now, obviously, this isn't just enough by itself because if I do negative one to the nth power, that's just going to alternate from negative one to positive one. So we're going to have to figure out how to increase these numbers by 2 each time. So what do I do? Remember, a lot of times you're going to have to multiply constants or different things like that. So can I just multiply this whole sequence by 2? Well, what happens here is if I just carry out this sequence and figure out the couple of numbers, I'm going to have negative 1, positive 1, negative 1, positive 1 times 2. So in other words, it's just going to be negative 2, positive 2, negative 2, positive 2. So that's not enough either. Now remember, whenever you increase by the same number each time, like plus 1 or plus 2 or plus 3, that means that you're going to have either \( n \) or \( 2n \) or \( 3n \) or so on. So in this case, what's going to happen is we're going to have to increase or going to have to multiply this by \( 2n \). Alright? So this \( 2n \) is now going to account for the fact that now for each term, you're going to increase by 2 as the index gets higher. So now let's take a look at the first couple of terms here and see if this sort of lines up with the terms that we have. So for \( a_1 \), this is going to be negative one to the one power times two times one. So in other words, this is just going to be negative one times 2, which is negative 2. That's exactly what the first term is. Let's take a look at the second term. This is going to be negative one to the second power times 2 times 2. So in other words, this is just going to be positive one times 4. That's going to give us positive 4. That's exactly what the second term is. One more just for, you know, just to be safe. This is going to be negative one to the third power times two times three. So in other words, this is going to be negative 1, right, because it's to the 3rd power. It's an odd power. And this is going to be times 6, so this is going to be negative 6. And that's exactly where our 3rd term is. So it turns out that this is the general formula for the sequence here, negative one to the nth power and then times \( 2n \). Now what's also really important about this is that you can't just sort of merge these two things in. So, for example, you can't say something like negative \( 2n \) to the nth power. It doesn't work like that. You can't like absorb something into that parentheses. Alright? So you're just going to have to leave it like this general formula. So now how would we figure out the 18th term? Very simple. We just plug in 18 for the \( n \) for the index in this formula. So the 18th term is just going to be negative one raised to the 18th power times 2 times 18. Now if you plug this into your calculator, what you should see is that negative one to the 18th power is just 1, right, because it's an even power, and then 1 times 36 just equals positive 36. So that's the general formula. And then for the 18th term, we just have positive 36. Alright? So thanks for watching. Hopefully, this made sense, and I'll see you in the next one.