Everyone, welcome back. So we just spent a lot of time talking about recursive formulas for arithmetic sequences, and then we saw how to write the general formula. Well, we just learned how to write a recursive formula for a geometric sequence that allows you to calculate terms based on the previous term. But now we're going to take a look at how to write the general formula, which allows you to calculate any term without having to know the previous one. And what we're going to see is there are a lot of parallels with how we did this for arithmetic sequences. I'm going to show you how to write a general formula for a geometric sequence, and then we'll do an example together. Let's get started here.
So, a general formula is going to give you the nth term. And just like the arithmetic sequence, it's going to be based on the first term. But now instead of the common difference \(d\), it's going to be based on the common ratio \(r\). Alright? So let's take a look here.
So with an arithmetic sequence, we saw that for a recursive formula, you're just doing \(a_{n-1} + d\). And then for the general formula, it's \( a_1 + d \cdot (n-1) \). Well, for a geometric sequence, it's going to be a little different. In this sequence, we saw that the common ratio between the terms, the \(r\) term, was equal to 2. You're multiplying by 2 each time. So in other words, this was the recursive formula. Well, and we saw that the sort of formats for any recursive formula is \( a_{n-1} \times r \). Well, for a general formula, again, your textbooks are going to do some derivations for this. I'm just going to show you the nth term, the general term, is going to be the first term times \(r\), but raised to a power of \(n-1\), not multiplied by \(n-1\).
So here, we can again see lots of similarities between these two types of equations. So for example, in this sequence, 3, 6, 12, 24, all you need to write the general formula is the first term, which is this 3 over here, and then you just need the common difference. And again, this common difference, what we see here is that \(r = 2\). These are the 2 things you need to write the general formula. \(a_n\) is just going to be the first term, which is 3 times the common difference, which is 2 raised to the \(n-1\) power.
This is the general formula for this sequence over here, whereas this was the recursive formula. Alright? Now, let's use this to actually calculate the 4th term, which we know should be 24. Well, \(a_4\), where \(n = 4\), it says that we're just going to take the first term, which is 3, and multiply it by 2 raised to the \(4-1\) power. In other words, this is \(3 \times 2^3\). This is \(3 \times 8\), and \(3 \times 8\) does, in fact, give us 24, exactly what we'd expect. Alright?
So this is the general sort of formats for a general formula. It's the first term times \(r\) raised to the \(n-1\) power. Alright? So when it comes to problems, and you're asked to write a formula for the general or nth term, you're always going to start with this equation over here. Alright. So let's take a look at our example. We're going to actually, we've actually taken a look at the sequence already, 5, 20, 80, 320. We're going to write a general formula for this, and we're going to use it to find the 12th term. Remember, finding really high index terms is always going to be a nightmare by using recursive formulas, so we want to use a general formula for this instead. Let's get started.
So if we want to write a general formula, we're just going to start out with our general formula equation, which is that \(a_n = a_1 \times r^{n-1}\). Alright? So all we need is \(a_1\), the first term, which actually is just the 5 that we're given right here. That's the 5. So this is \(a_1\). And we also need the common ratio. We've already seen the sequence before. Notice how each one of these terms actually gets multiplied by 4 to get to the next term. So, in other words, this is the common ratio, \(r = 4\).
Alright? So we have what our two terms are, \(a_1\) and \(r\). So this is just going to be \(a_1\) equals well, our first term is 5 times our common ratio, which is 4 raised to the \(n-1\) power. So this is all we need. That's our general formula for this sequence. Let's go ahead and take a look at this general formula and use it to calculate the 12th term. We just plug in \(n = 12\) in for this formula. What we're going to see here is that it's 5 times 4 raised to the \(12-1\) power. And, so, in other words, this is going to be 5 times \(4^{11}\). Now, obviously, you can use your calculator for this. This is going to be a huge number. And what it turns out to be is it turns out to be 20,971,520. This is a huge term, but this would be the 12th term in this sequence. You're obviously going to have to use calculators to solve this, but that's all there is to it. So we can see here that the structure of this is very similar to arithmetic sequences. This is the formula for a general formula for a geometric sequence. Let me know if you have any questions.