Geometric Sequences - Online Tutor, Practice Problems & Exam Prep

Geometric sequences feature a constant ratio between consecutive terms, known as the common ratio (r). The recursive formula is expressed as an = an-1⁢r, while the general formula is an = a1⁢r⁢n-1. Understanding these formulas allows for efficient calculation of any term in the sequence, highlighting the rapid growth characteristic of geometric progressions compared to arithmetic sequences.

Geometric Sequences - Recursive Formula

Everyone, welcome back. So we just finished talking a lot about arithmetic sequences, like, for example, 3, 6, 9, 12, where the difference between each number is always the same number. But let's take a look at this sequence over here, 39, 278, 1. 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 going to show you in this video is that this is a special type of sequence called a geometric sequence, and what we're going to see is that there are 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 want to 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 that the sequence was 3. 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 3. From 9 to 27, you also multiply by 3. From 27 to 81, you multiply by 3. So, instead of adding 3 to each number to get the next one, you have to multiply by 3 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 3. Kind of like how in this case little d was equal to 3. Alright?

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 3. Well, in this geometric sequence, we're going to take the previous term, and instead, we have to multiply by 3. That's really all there is to it. The way that you use these formulas to find the next terms is exactly the same. Alright?

So, in fact, this sort of general sort of structure that you'll see for these recursive formulas for geometric sequences is they'll always look like this: A_n, the new term, is going to be the previous term times r, whatever that common ratio is. Alright?

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 the next term. And what we're going to see here also is that, 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, 278, 1, whereas this is only 3, 6, 9, 12. So these tend to grow much faster than arithmetic sequences. Alright?

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, 80, and 320. 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 5 to get to 20? Well, the first thing you're going to do here is you're going to find r by dividing any 2 consecutive terms. So what we're going to 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 4. From 20 to 80, you also multiply by 4. And from 80 to 320, you also multiply by 4. So, clearly, in this case, our r, our common ratio, is equal to 4. Alright?

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 going to be something that looks like A_n equals A_n-1, the previous term, times r. In other words, times r, a common ratio. So this is just going to be A_n-1 times 4. Alright?

Now, remember, 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, A₁ is equal to 5. Alright?

So that's how to write recursive formulas. And to find the next one, you would just take the previous term and multiply by 4. Alright? 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

Welcome back, everyone. 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's 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 \(a_1 (the first term) + d(n-1)\). Well, for a geometric sequence, it's going to be a little bit 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, we saw that the sort of formats for any recursive formula is \(a_{n-1} \times r\). Well, for a general formula, 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 the power of \(n-1\), not multiplied by \(n-1\). So here we can again see lots of similarities between these two types of equations.

For example, in this sequence, 3, 6, 12, 24, all you need to write the general formula is the first term, which is 3 over here, and then you just need the common ratio. And again, this common ratio, what we see here is that \(r = 2\). These are the two things you need to write the general formula. \(a_n\) is just going to be the first term, which is 3 times the common ratio, 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 to the 3rd power. 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 to the 11th power. 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.

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

• A. 5120
• B. 1280
• C. 10240
• D. 2560

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. $a_{n}=\frac23\cdot\left(-3\right)^{n-1}$

Here’s what students ask on this topic:

What is a geometric sequence and how is it different from an arithmetic sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio (r). In contrast, an arithmetic sequence is a sequence where each term is found by adding a constant difference (d) to the previous term. For example, in the geometric sequence 3, 9, 27, 81, the common ratio is 3, while in the arithmetic sequence 3, 6, 9, 12, the common difference is 3. Geometric sequences grow exponentially, whereas arithmetic sequences grow linearly.

How do you find the common ratio in a geometric sequence?

To find the common ratio (r) in a geometric sequence, divide any term by the previous term. For example, in the sequence 5, 20, 80, 320, you can find the common ratio by dividing 20 by 5, which gives 4. Similarly, dividing 80 by 20 also gives 4, confirming that the common ratio is consistent throughout the sequence. Mathematically, this can be expressed as $\frac{\mathrm{a_n}}{\mathrm{a_n-1}}$ = r.

What is the recursive formula for a geometric sequence?

The recursive formula for a geometric sequence is used to find the next term based on the previous term. It is expressed as $\mathrm{a_n}=\mathrm{a_n-1}*r$, where $\mathrm{a_n}$ is the nth term, $\mathrm{a_n-1}$ is the (n-1)th term, and $r$ is the common ratio. For example, if the first term $\mathrm{a₁}$ is 5 and the common ratio $r$ is 4, the sequence would be 5, 20, 80, 320, etc.

How do you write the general formula for a geometric sequence?

The general formula for a geometric sequence allows you to find any term without knowing the previous term. It is expressed as $\mathrm{a_n}=\mathrm{a₁}*rn-1$, where $\mathrm{a_n}$ is the nth term, $\mathrm{a₁}$ is the first term, $r$ is the common ratio, and $n$ is the term number. For example, if the first term $\mathrm{a₁}$ is 3 and the common ratio $r$ is 2, the nth term is given by $\mathrm{a_n}=\; 3\; *\; 2n-1$.

How do you find the nth term of a geometric sequence?

To find the nth term of a geometric sequence, use the general formula $\mathrm{a_n}=\mathrm{a₁}*rn-1$. For example, if the first term $\mathrm{a₁}$ is 5 and the common ratio $r$ is 4, to find the 12th term, substitute $n$ with 12: $\mathrm{a₁₂}=\; 5\; *\; 411$. This results in $\mathrm{a₁₂}=\; 5\; *\; 4194304\; =\; 20971520$.