Arithmetic Sequences - Online Tutor, Practice Problems & Exam Prep

Arithmetic sequences are characterized by a constant difference, known as the common difference (d), between consecutive terms. A recursive formula for an arithmetic sequence can be expressed as an = an-1+d, where the first term is specified. The general formula for the nth term is an = a1+d(n-1), allowing for efficient term calculation without prior terms. Understanding these formulas is essential for analyzing sequences effectively.

Arithmetic Sequences - Recursive Formula

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 a₁, which is equal to 3. And I want to calculate a₂. So a₂ says that I'm going to take the previous term in the sequence, a₁, 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, a₃ is going to be if I take the previous term a₂ 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 a₄. This is going to be a₃ 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 a₁ 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. a₃ says that this is going to be a₂ minus 6. I already know what a₂ is. It's equal to 3, so 3 minus 6 is negative 3. And a₄ is just going to be a₃, 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.

Write a recursive formula for the arithmetic sequence.

$\left\lbrace8,2,-4,-10,\ldots\right\rbrace$

• A. $a_{n}=a_{n-1}-10$ ; $a_1=6$
• B. $a_{n}=a_{n-1}-10$ ; $a_1=6$
• C. $a_{n}=a_{n-1}-6$ ; $a_1=8$
• D. $a_{n}=a_{n-1}-10$ ; $a_1=8$

Arithmetic Sequences - General Formula

Everyone, welcome back. So in previous videos, we saw how to write recursive formulas for arithmetic sequences. For example, in this specific sequence, we saw this was the recursive formula: You take the previous term and then add 5 to it. But in some problems, you may be asked to write a formula for the general or the nth term in a sequence, and you may have to use it to find something like the 101st term in a sequence, something with an incredibly high index. Doing this with a recursive formula would be a nightmare because you'd have to calculate first the first one hundred terms. So instead, what I'm going to do in this video is show you how to write a general formula for this arithmetic sequence. Remember, these equations allow you to calculate any terms without having to know what the previous terms are. So I want to go ahead and show you the difference between these two, and we'll do some examples together. Let's get started.

General formulas are always formulas that contain n, whereas recursive formulas always contain a_n-1, the previous term in a sequence. Well, for a general formula for arithmetic sequences, it'll give you the nth term, and it's based on a couple of things. It's based on the first term in the sequence and the common difference, d, that lowercase d we've been working with. So your textbooks are going to do some lengthy derivations for this. I'm actually just going to go ahead and give you what the formula looks like. To calculate the nth term in a sequence, you're going to have to know the first term, a₁, that's what we just said, plus d times n-1. I want to go ahead and make sense of this equation using the sequence that we have over here, 2, 7, 12, 17, so on and so forth. Remember, these are the first four terms in the sequence, a_{1, 2, 3, 4}, so on and so forth. So notice how the difference between a₁ and a₂ to get from a₁ to a₂, you just have to add one multiple of d, the common difference. Now to get from 2 to 12, you have to go and start at the first term, and then you have to add 2 times d, that multiple or that common difference. And to get to the 4th term, you have to know what the first term is, and then you have to add 3 times d. So notice how there's a pattern going on here. To calculate the nth term, you always start at the first term, and then you have to multiply d by n-1. That's what's going on here in this formula. So for this specific sequence over here, the general formula would just be the first term in the sequence, which in this case is 2, plus the common difference, which we know from this sequence over here that it's 5, and then we multiply it by n-1. This is the general formula for this sequence over here. We can use it to calculate any terms, 2, 3, 4, whatever. Let's go ahead and calculate the 4th term of the sequence using this formula because we know from the recursive formula that it should be 17. So we're just going to sort of double-check here. So a₄ would be a₁, which is 2, plus 5 times n - 1, which is going to be 4 - 1. Remember, this is just going to be where n=4, so we're going to have to plug in 4 - 1. So this is just going to be 2 + 5×3, in which case you're going to get, 2 + 15, which is 17. And that's exactly what we should get here, what we should expect. So using this general formula, we found that the 4th term is 17. Alright? That's really all that's going on here. This is always going to be the formula that you start off with when you're writing a general formula for arithmetic sequences.

Let's go ahead and take a look at another example and work this one out together. So for this sequence below, we're going to write a formula for the general or nth term. So this just means we're going to write a general formula and use it to find the 101st term. So we have the sequence over here. I've got 2, 5, 8, 11, and 14. So let's just go ahead and write out our general formula equation, a_n = a₁ + d*n-1. Now it might seem kinda scary at first because it's lots of letters that are going on here. But, really, all you need to know is you need to know what a₁ is and d, that common difference. These are the only numbers you're going to plug in for because, remember, n is going to be your index. You don't plug in for that unless you're actually finding a certain term. Alright? So what is a₁? Well, that's pretty easy because you just look at the first term over here. So this is just going to be a₁. That's going to be the 2. Alright? So in other words, your formula is just going to be 2 plus now what's the common difference in this? Well, you may just take a look at the numbers, 2, 5, 8, 11, 14. There's a pattern going on, which is that each one of these things increments by 3. And that's going to be that common difference. That's what d equals. It's positive 3. So this is going to be 2 + 3. This is your general formula for this arithmetic sequence. That's all there is to it. You don't plug anything in for n just yet because this just shows you how to calculate the general term a_n. All right? So now that we have this general formula, now we can use it just to find the 101st term. All we have to do is just plug in 101 in for n inside of this equation. So what this says here is that we're going to take the first term, which is 2, and we're going to add the common difference of 3 times 101 minus 1. In other words, this is going to be 2 + 3×100. And so, in other words, the 101st term is really just going to be 3 times 100 plus 2, which is 302. Very easy. So if we had to do this via recursive formula, it would have taken us forever to do this. And that's why these general formulas are super powerful. Anyway, thanks for watching, folks. That's it for general formulas. Let's go ahead and get some practice.

Find the general formula for the arithmetic sequence below. Without using a recursive formula, calculate the $30^{\th}$ term.

$\left\lbrace-9,-4,1,6,\ldots\right\rbrace$

• A. 46
• B. 136
• C. 146
• D. 150

Example 1

Hey, everyone. So sometimes in some problems, you may be asked to write a general formula when you're given a recursive formula, like the problem we're going to work out down here in this example. We're going to use this to calculate the 15th term in this sequence. Remember, trying to do this just recursively is going to take forever. That's why these general formulas are really powerful. So let's take a look here. We already have a recursive formula for an arithmetic sequence, where the next term is just the previous term plus 3. We have the starting term, which is 2. That's what we need for an arithmetic sequence. So how do we actually get the general formula from this? It's actually really straightforward. Remember, all you need for a general formula is you just need the first number in the sequence, which is actually already given to us, and then you just need the common difference, lowercase d. And then you just multiply it by the index minus 1, n minus 1. That's all you need for the general formula. So if you're given the recursive formula for this, remember that this number that goes in front of the previous sequence or after the previous sequence, this is d. This is your common difference d. So we actually already have the two numbers that we need for our general formula, and we can sort of convert one formula into another. It's very straightforward. So if I can pull both of these things together into our formula here, what this says is that the nth term in the sequence is going to be the first term plus the common difference times n minus 1. Alright? So in other words, I'm just going to take that first number of the sequence, which I already know is 2, that's going to be 2, and I'm going to take the common difference, which I know is 3, and I'm going to multiply this by the index n minus 1. Alright? And that's going to be my nth term. This is the general formula for the sequence over here. In fact, you can use this to calculate the first two terms, and what you'll see here is that a subscript 2 is equal to 5, a subscript 3 is equal to 8, a subscript 4 is equal to 11, just as we would get with a recursive formula. Alright? So that's the general formula. It's just the first term plus the common difference times n minus 1.

Now let's use this to calculate the 15th term in the sequence. So the 15th term of the sequence is just where n equals 15, and you just plug that into your formula. The first term in the sequence is 2 + 3 times, and then the only place where you plug in the n is in the n minus 1. So this is going to be 15 minus 1. So in other words, this is just going to be 2+3×14, and 2+3×14 gives us a 15th term of 44. So that's the general formula, and the 15th term of the sequence is 44. Pretty straightforward. Alright? Thanks Thanks for watching. Let me know if you have any questions. Thanks. We'll see you in the next one.

Example 2

Everyone, welcome back. So, I hope you'll get a chance to try out this example problem. And if you got a little stuck and didn't know where to go, that's perfectly okay. This one's a little bit of a tricky one. Let's go ahead and get started here. The 4th and 6th terms of a sequence are given to us. We're told that \( a_4 \) is negative 2 and \( a_6 \) is equal to 6. And we're supposed to use this information to find the 18th term of the sequence. Now remember, whenever we're trying to find really high indexes of sequences, you're always going to want to start with a general formula. You don't want to do it recursively. Right? So let's start out there. So we're told that \( a_n \) is going to be \( a_1 + d(n - 1) \). That's the general formula. So do we know \( a_1 \) and \( d \), and can we figure it out from the information that is given to us? Well, not really, because the only thing that we're told here is we're actually just given two numbers in the sequence, and none of them are \( a_1 \). We're told that \( a_4 \) is negative 2 and \( a_6 \) is 6. So in other words, I don't know what the first term in this sequence is, and because I don't even have 2 consecutive terms, \( a_4 \) and \( a_6 \), I don't even know what the common difference is between these terms either. So how do I even get started with this problem? Well, let's just actually use the information that we do know in this example. So from this general formula here, what we're told is that \( a_4 \) is equal to \( a_1 + 3d \). And if we're using \( n = 4 \) here, then this is just going to be 3. \( a_6 \) is just going to be \( a_1 + 5d \). And if we're using \( n - 1 \) and \( n = 6 \), this is going to be 5. Now we actually know what these numbers equal. This is equal to negative 2, and this is equal to 6. So how do I use this information over here to find any one of the variables that I need to know? I need to know \( a_1 \), and I need to know \( d \). So now if you notice here, what I've ended up with is I've ended up with a situation where I have 2 equations. Right? I've got these 2 equations in which these two variables, \( a_1 \) and \( d \), are unknown variables. I've got 2 equations with 2 unknown variables. The whole idea here is that we're going to turn this into a system of equations. So we've got \( a_1 + 3d \) is equal to negative 2, and this is going to be \( a_1 + 5d \) is going to be 6. That's what my two equations say. Right? I just rewrote these 2 equations over here. The whole idea is that this is basically just a system of equations. And so back when we studied systems of equations, we saw different methods to solve these types of problems. We're going to use that here. This sort of system of equations is kind of like so I'm going to put this like when you had a situation where you had something like \( x + 3y \) is equal to negative 2, and \( x + 5y \) is equal to 6. So remember when we had \( x \)'s and \( y \)'s and we used different methods to solve these types of problems? Well, if you already have their coefficients lined up, then we could use here we could use the elimination method, where we basically subtract 2 equations to get rid of 1 of the variables and isolate one of them, and then we can use one equation with 1 variable, and that was easier to solve. That's exactly what we're going to do here. We're going to take these 2 equations here, and we're going to subtract them. So remember, this is the elimination method, so this is elim. Alright? So what I'm going to do here is we're basically going to subtract out the first terms, the \( a_1 \) terms over here, because those coefficients are the same. They're equal but opposite. Alright? And so when you subtract these two equations, what do you get? Well, remember, you're going to sort of subtract everything vertically straight down. Right? So you subtract all these terms. \( 3d - 5d \) gives me \( -2d \), and then \( -2 - 6 \) gives me \( -8 \). So if you divide by \( -2 \) on both sides, what you'll end up with is that \( d \) is equal to 4. So without knowing the general formula, without even knowing the sequence, just knowing 2 of the numbers, I was able to figure out the common difference is equal to 4. And if you think about this, this actually kind of does make sense because if \( a_4 \) is equal to negative 2, then that means that \( a_5 \), the next term in the sequence, would be 2. And now this totally makes sense because from 2 to negative sorry. Negative 2 to positive 2, that's a difference of 4. From 4 to 6, that's another difference of 4. Alright? So the common difference in the sequence is definitely 4. So that's one of our numbers now. We actually have the common difference \( d \), and now we just need to figure out, well, what's \( a_1 \) in this general formula? So we've got \( d \), and we need to figure out \( a_1 \). Well, just as how we use a system of equations where once we found one variable, you could plug it back into either one of these equations to solve for the other. We can do the exact same thing here. I can take this \( d \) and plug it into either one of these two equations, the two equations that I actually do know numbers for, to figure out what the first term of the sequence is. It doesn't matter which one you pick. Feel free to choose whichever one you want. And I'm just going to go ahead and pick the one with the slightly smaller numbers. So I know that \( a_1 + 3d \) over here is going to equal negative 2. Alright? So what I'm going to do here is I'm just basically going to replace \( d \) with 4 so I can figure out \( a_1 \). So what it says here is that \( a_1 + 3 \times 4 \) is going to equal negative 2. So this is going to be \( a_1 + 12 \) equals negative 2, and I can subtract 12 from both sides. What I'll see here is that \( a_1 \) is equal to \( -14 \). So that is the first term in my sequence. And, again, I figured this out by knowing what the common difference is and then plugging it back into one of my two terms that I actually do know. Alright? So that's one of my terms, and that's the other one that I need for the general formula. So now I can actually write out my general formula over here. This is going to be my general formula, and what this says is that the \( n \)th term is going to be the first term, \( -14 \) plus \( d \), in which case this is \( 4 \times (n - 1) \). So this is the general formula over here. I'm going to write this out. This is the general formula. Now, how do we use this to find out the 18th term? Well, actually, this is pretty straightforward. Now, we're just going to use \( a_n \) over here, is going to be \( -14 + 4 \times (17) \). So, if you do \( -14 + 4 \times 17 \), then what you're going to get here is you're going to get an answer of 54. So that means that the 18th term I'm sorry. I actually meant to write 18 over here. The 18th term is going to be positive 54. Alright? So that is the 18th term in the sequence. That's how you take these types of problems where you know 2 terms. You need to write a general formula. It really actually just turns into a system of equations. And then from there, you can solve. Thanks so much for watching. Hopefully, you got this. And if not, that's okay. It's a little bit tricky. Let's go ahead and move on.

Here’s what students ask on this topic:

What is an arithmetic sequence and how do you identify it?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by $d$. To identify an arithmetic sequence, check if the difference between each pair of consecutive terms is the same. For example, in the sequence 2, 5, 8, 11, the common difference is 3 because each term increases by 3.

How do you write a recursive formula for an arithmetic sequence?

To write a recursive formula for an arithmetic sequence, use the format $a_n=a_{\mathrm{n-1}}+d$, where $a_n$ is the nth term, $a_{\mathrm{n-1}}$ is the previous term, and $d$ is the common difference. Additionally, specify the first term $a_1$. For example, for the sequence 3, 7, 11, 15 with a common difference of 4, the recursive formula is $a_n=a_{\mathrm{n-1}}+\; 4$ with $a_1=\; 3$.

How do you find the nth term of an arithmetic sequence using the general formula?

To find the nth term of an arithmetic sequence, use the general formula $a_n=a_1+d(n-\; 1)$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. For example, in the sequence 2, 7, 12, 17 with $a_1=\; 2$ and $d=\; 5$, the 4th term is calculated as $a_4=\; 2\; +\; 5(4\; -\; 1)\; =\; 2\; +\; 15\; =\; 17$.

What is the common difference in an arithmetic sequence and how do you find it?

The common difference in an arithmetic sequence is the constant amount by which each term increases or decreases. It is denoted by $d$. To find the common difference, subtract any term from the subsequent term. For example, in the sequence 4, 9, 14, 19, the common difference is $9\; -\; 4\; =\; 5$ or $14\; -\; 9\; =\; 5$. The common difference remains the same throughout the sequence.

How do you determine if a sequence is arithmetic?

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. This difference is called the common difference, $d$. For example, in the sequence 10, 15, 20, 25, the difference between each pair of consecutive terms is $15\; -\; 10\; =\; 5$, $20\; -\; 15\; =\; 5$, and $25\; -\; 20\; =\; 5$. Since the difference is constant, the sequence is arithmetic.