Geometric Sequences - Video Tutorials & Practice Problems

Geometric Sequences - Recursive Formula

Hey everyone, welcome back. So we just finished talking a lot about arithmetic sequences like for example, 369 12, where the difference between each number is always the same number. But let's take a look at this sequence over here. 39 27 81. Clearly, we can see that the difference between numbers is never the same. It's constantly getting bigger, but there's still actually a pattern going on with this sequence. What I'm gonna show you in this video is that this is a special type of sequence called a geometric sequence sequence. And what we're gonna see is that there's a lot of similarities between how we use the information and the pattern across the numbers to set up a recursive formula for these types of sequences. So I wanna show you how to do that and the basic difference between these two types. And we'll do some examples. Let's get started. So remember that arithmetic sequences are special types where the difference between terms was always the same. For example, the common difference in this situation of the sequence was three, a geometric sequence is a special type where the ratio between terms is always the same number. So for example, from 3 to 9, you have to multiply by three, from 9 to 27 you also multiply by three from 27 to 81 you multiply by three. So instead of adding three to each number to get the next one, you have to multiply by three to get the next number. Now this ratio over here is called the common ratio. And the letter we use for this is little R. So little R in this case is equal to three kind of like how in this case little D was equal to three. All right. Now, we can use this common ratio to find additional terms by setting up a recursive formula. Remember, recursive formulas are just formulas that tell you the next term based on the previous term. So in this situation, we just took the previous term and added three. While in this geometric sequence, we're gonna take the previous term and instead we have to multiply by three. That's really all there is to it, the way that you use these formulas to find the next terms is exactly the same. All right. So and in fact the sort of general sort of a sort of a structure that you'll see for these recursive formulas for geometric sequences is they'll always look like this. A and the new term is gonna be the previous term times R whatever that common ratio is. All right. So clearly we can see here that the only difference between these two is the operation that's involved for arithmetic, you always add numbers to get to the next term. Whereas in geometric sequences, you multiply numbers to get with the next term. And what we're gonna see here also is that uh generally addition grows a little bit slower than multiplication. So these types of sequences, the numbers grow a little bit slower. Whereas in geometric sequences, these tend to grow very fast because they're exponential, right, they're going 39, 2781 whereas this is only 369 12. So these tend to grow a much faster and arithmetic sequences. All right. So let's go ahead and take a look at our example here because sometimes you may be asked to write recursive formulas for geometric sequences. And in fact, there's a lot of similarities between how we did this for arithmetic sequences. All you have to do is first find the common ratio and then we can set up using this equation over here. Let's get started with this example. So we have the numbers 5 20 8300 and 20. Notice how the difference is not the same between the numbers, but there is a pattern that's going on here. So what do I have to do to five to get to 20? Well, the first thing you're gonna do here is you're gonna find R by dividing any two consecutive terms So we're gonna have to do is take a look at the pattern between the two numbers. And what we can see here is that from 5 to 20 you have to multiply by four, from 20 to 80 you also multiply by four and from 80 to 320 you also multiply by four. So clearly, in this case, our r our common ratio is equal to four. All right. So now we just use this, we move on to the second step, which is we're going to write a recursive formula. A recursive formula is gonna be something that looks like A N equals A and minus one. The previous term times R. In other words, times are a common ratio. So this is just gonna be A and minus one times four. All right. Now remember just like in uh just like for arithmetic sequences, having this formula by itself isn't useful or isn't helpful because you have to know what the first term is. So you always have to write what the first term in the sequence is. In this case, a one is equal to five. All right. So that's how to write recursive formulas and to find the next one, you would just take the previous term and multiply by four. All right. So that's it for this one. Folks. Let me know if you have any questions. Thanks for watching.

Write a recursive formula for the geometric sequence $\left\lbrace18,6,2,\frac23,\ldots\right\rbrace$.

• A. $a_{n}=3a_{n-1}$
• B. $a_{n}=\frac{a_{n-1}}{3}$
• C. $a_{n}=18a_{n-1}$
• D. $a_{n}=\frac23a_{n-1}$

Geometric Sequences - General Formula

Hey, 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 gonna see there's a lot of parallels with how we did this for arithmetic sequences. I'm gonna 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 gonna give you the nth term and just like the arithmetic sequence is gonna be based on the first term. But now instead of the common difference D, it's gonna be based on the common ratio R, all right. 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 minus one plus D. And then for the general formula, it's a one, the first term plus D multiplied by N minus one. Well, for a geometric sequence, it's gonna be a little bit different in this sequence. We saw that the common ratio between the terms, the R term was equal to two, you're multiplying by two 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 and minus one times R. Well, for a general formula, again, your textbooks are gonna do some derivations for this. I'm just gonna show you the A nth term. The general term is gonna be the first term times R but raised to a power of N minus one, not multiplied by N minus one. So here we can again see lots of similarities between these two types of equations. So for the, for example, in this sequence, 36, 1224 all you need to write the general formula is the first term which is this three over here. And then you just need the common difference. And again, this common difference, what we see here is that R equals two, these are the two things you need to write the general formula A N. It's just gonna be the first term which is three times the common difference, which is two raised to the N minus one power. This is the general formula for this sequence over here. Whereas this was the recursive formula. All right. Now, let's use this to actually calculate the fourth term which we know should be 24 A four where N equals four. It says that we're just gonna take the first term we, which is three and multiplied by two, raised to the four minus one power. In other words, this is three times two to the third power. This is three times eight and three times eight does, in fact give us 24 exactly what we'd expect. All right. So this is the general sort of format for a general formula. It's the first term times R raised to the N minus one power. All right. So when it comes to problems and you're asked to write a gene a formula for the general or nth term, you're always gonna start with this equation over here. All right. So let's take a look at our example. We're gonna actually, uh we've actually taken a look at the sequins already 5, 20 8300 and 20. We're gonna write a general formula for this and we're gonna use it to find the 12th term. Remember finding really high index terms is always gonna 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 gonna start out with our general formula equation, which is that A N equals a one times R to the N minus one power. All right. So all we need is we need a one, the first term, which actually is just the five that we're given right here. That's the five. So this is a one and we also need the common ratio. We've already seen this sequence before notice how each one of these terms actually gets multiplied by four to get to the next term. So in other words, this is the common ratio R equals four. All right. So we have what our two terms are uh A one and R. So this is just gonna be a one equals while our first term is five times our common ratio, which is four raised to the N minus one power. So this is all we need. That's our general formula for the 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 and equal 12 in for this formula. What we're gonna see here is that it's five times four raised to the 12 minus one power. And so in other words, this is gonna be five times four to the 11th power. Now, obviously, you can use your calculator for this. This is gonna be a huge number. And what it turns out to be is it turns out to be 20 million, 917,000 uh 500. 0, I'm sorry, 1,042,520. This is a huge term, but this would be the 12th term in this sequence. You're obviously gonna 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.

Find the $10^{\th}$ term of the geometric sequence in which $a_1=5$ and $r=2$.

• A. 5,120
• B. 1,280
• C. 10,240
• D. 2,560

Write a formula for the general or $n^{\th}$ term of the geometric sequence where $a_7=1458$ and $r=-3$.

• A. $a_{n}=1\cdot\left(-3\right)^{n-1}$
• B. $a_{n}=2\cdot\left(-3\right)^{n-1}$
• C. $a_{n}=-\frac23\cdot\left(-3\right)^{n-1}$
• D. $a_{n}=\frac23\cdot\left(-3\right)^{n-1}$