Arithmetic Sequences - Video Tutorials & Practice Problems

Arithmetic Sequences - Recursive Formula

Hi, everyone. Welcome back. So throughout our discussion on sequences, we've often run across ones that look like this with numbers 27, 1217, where the numbers increase by the same amount each time. In this case, each one of these numbers increases by five. 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. Well, we're gonna 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 gonna show you how to write recursive formulas for them turns out that it is a very straightforward formula that we'll see that only just depends really on one variable. So I wanna 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 gonna be the same number. So notice how in this sequence over here 27 1217, the first term in the sequence is two and then each term after that increases by five, the difference between any number and the previous number is always five. 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 five, 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 gonna set up a recursive formula which remember says that the next term is the previous term and then we're gonna 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 five to it. So recursive formula for this sequence is just that A N equals A N minus one, the previous term, but you just have to add five to it. That's really all there is to it. This is the recursive formula that tells us the next few terms in this sequence. All right. Now, all we have to do is just specify what the first term in the sequence is, which in this case is two. But that's really how you set up a recursive formula. So the general sort of format for this is that you're gonna have the, the new term is equal to the previous term plus D, whatever that common difference is in this case it was equal to five. That's really all there is to it, all of your arithmetic sequences will always have recursive formulas that look like this. All right, let's go ahead and take a look at our first example. In this example, we're actually given what the recursive formulas are. So over here, an example A and B we want to do is we want to write the first four terms. All right, let's get started. So we actually already have what the first term is, which is a one which is uh equal to three. And I wanna do calculate a two. So a two says that I'm gonna take the previous term in the sequence A one and I'm gonna have to add four to it. And if I do that, that's just gonna be seven, right? Because I have three plus four and this is gonna equal seven. So no, a three is gonna be if I take the previous term A two and then add four to it. So in this case, I just calculated what that is, that's just seven. So that's gonna be seven plus four and this is gonna give me 11 and I can do the same thing for a four. This is gonna be a three plus four, which in this case is just gonna be 11. So it's gonna be 11 and then plus four which will give me 15. So it turns out that these are the first four terms in the sequence. We got 3, 7-Eleven and 14. All right. So I just have to take the previous term and add four onto it. All right. So let's take a look. Now at the second example here, we have the recursive formula that says that the new term is the previous term. But now we're gonna subtract six and we're gonna start off with the first term being nine. So let's take a look here. So a one is equal to nine. I wanted to calculate the second term with this formula says is I'm gonna have to take the first term and then subtract six. But I know what the first term is. It's nine. So nine minus six is equal to three. Let's take a look at the third term. A three says that this is gonna be a two minus six. I already know what A two is equal to three. So three minus six is negative three and a four is just gonna be a three which I just calculated was negative three minus six. So this is gonna be negative three minus six and this gives me negative nine. So notice how each one of these numbers is continuously decreasing by six each time. So sometimes your common difference can be a positive number like plus four and sometimes it could be a negative number like minus six. That's perfectly fine here, right? So here D equals negative six here D equals four and you can still calculate the next few terms in the sequence. All right. 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 ter a sequence of terms and whatever you were trying to write a recursive formula, we're first gonna have to find that common difference. That lower case D let me show you how to do this step by step here. So in this case, we're gonna write a recursive formula for the sequence. 258, 1114. Notice how this is definitely an arithmetic sequence because the difference between each term and the next one is always the same. Notice how we're just adding three to each previous term to get the new one. So what you're gonna do here is to write a recursive formula, remember that it's gonna look something like this, right? So your A N is just gonna be A N minus one plus the common difference. So the first step is actually to find that common difference by subtracting any two consecutive terms. And really, it doesn't matter if you do five minus two or eight minus five or 11 minus eight, it's always gonna 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 gonna get sort of a negative number, right? In this case, what you're gonna do here is every term is adding three. So that means D is equal to three. In other words, five minus two is equal to positive three, right? So that's D it's three. 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 three. 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 three 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, that starts off the because otherwise you haven't, you don't know where to start, right. So this is a, an equals A and minus one plus three. And the first term that you start off with is gonna be the number two. Using these two pieces of information. I could recreate any term in this sequence over here. All right. 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. So for example, in this specific sequence, we saw this was the recursive formula, you take the previous term and then add five to it. But in some problems, you may be asked to write a formula for the general or the nth term in a sequence or you and you may have to use it to find something like the 101st term 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 1st 100 terms. So instead of what I'm gonna do in this video 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 wanna go 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 and minus one, 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 gonna do some lengthy derivations for this. I'm actually just gonna go ahead and give you what the formula looks like to calculate the nth term in a sequence. You're gonna have to know the first term. That's a one, that's what we just said plus D times N minus one. I'm gonna go ahead and make sort of uh show you how to make sense of this equation using the sequence that we have over here. +27, 1217, so on and so forth. Remember that these are the first four terms in the sequence A +1234, so on and so forth. So notice how the difference between a one and a two to get from a one to a two, 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 two times D that is that uh multiple or, or that common difference. And to get from the, then you get to the fourth term, you have to know what the first term is and then you have to add three times D. So notice how that there's a pattern going on here to calculate the A nth term, you always start at the first term and then you have to multiply D by N minus one that index subtracted by one. That's what's going on here in this formula. So for this specific sequence over here, the general formula be the first term in the sequence which in this case is two plus the common difference, which we know from this sequence over here that it's five. And then we multiply by N minus one. This is the general formula for this sequence over here. We can use it to calculate any terms 234, whatever. Let's go ahead and calculate the fourth term of the sequence using this formula because we know from the recursive formula that it should be 17. So we're just gonna sort of double check here. So a four would be a one which is two plus five times N minus one, which is gonna be four minus one. However, this is just gonna be where N equals four. So we're gonna have to plug in four minus one. So this is just gonna be two plus five times three. In which case you're gonna get uh two plus 15, which is 17. And that's exactly what we should get here, which we should expect. So using this general formula, we found that the, 1/4 term is 17. All right, that's really all that's going on here. This is always gonna 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 the sequence below, we're gonna write a formula for the general or N term. So this just means we're gonna write a general formula and use it to find the 101st term. So we have the sequence over here. I've got 258 and 14. So let's just go ahead and write out our general formula equation A N equals a one plus DN minus one. Now, it might seem kind of scary at first because it's a lots of letters that are going on here. But really all you need to know is you need to know what A one is and D that common difference, these are the only numbers you're gonna plug in for because remember N is gonna be your index. You don't plug in for that unless you're actually finding a certain term. All right. So what is a one? Well, that's pretty easy because you just look at the first term over here. So this is just gonna be a one that's gonna be the two. All right. So in other words, your formula is just gonna be two plus. Now, what's the common difference in this? Well, you may just take a look at the numbers 258, 1114. There's a pattern going on, which is that each one of these things increments by three. And so that's gonna be that common difference. That would, that's what D equals, it's positive three. So this is gonna be two plus three times N minus one. 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 sub N. All right. So now that we have this general formula, now we can use it this to find the A 101 term or 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 gonna take the first term which is two and we're gonna add the common difference of three times 101 minus one. In other words, this is gonna be two plus three times 100. And so, in other words, the 101 term is really just gonna be three times 100 plus two, which is 302. Very easy. So if we had to do this via recursive formula, it would have taken us forever to do this. That's why these general formulas are super powerful. Anyway, thanks for watching folks. That's it for uh 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

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 gonna work out down here. In this example, we're gonna use this to calculate the 15th term in this sequence. Remember trying to do this, just recursive is gonna 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 three, we have the starting term, which is two. 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 of minus one and minus one. 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 I can pull both of these things together into our formula here. What this says is that the nth term in the sequence is gonna be the first term plus the common difference times N minus one. All right. So in other words, I'm just gonna take that first number of the sequence which I already know is two, that's gonna be two. And I'm gonna take the common difference which I know is three. And I'm gonna multiply this by the index and minus one. All right. And that's gonna 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 two is equal to five, a three is equal to eight. A four is equal to 11 just as we would get with a recursive uh with a recursive formula. All right. So that's the general formula. It's just the first term plus the common difference times N minus one. 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 two plus three times and then the only where you plug in the end is in the N minus one. So this is gonna be 15 minus one. So in other words, this is just gonna be two plus three times 14 and two plus three times 14 gives us 1/15 term of 44. So that's the general formula. And the 15th term of the sequence is 44. Pretty straightforward. All right. Thanks for watching. Let me know if you have any questions. Thanks. Uh We'll see you in the next one.

Example 2

Hey, everyone. Welcome back. So, uh 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 OK. This one is 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 four is negative two and A six is equal to six. And we're supposed to use this information to find the 18th term of the sequence. Now, remember when, whenever we're trying to find really high indexes of sequences, you're always gonna wanna start with a general formula. You don't want to do recursive, right? So let's start out there. So we're told that the A N is gonna be a one plus D A and minus one. That's the general formula. So do we know a one and D and can we figure it out from the information that's 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 one we're told the A four is negative two and A six is six. So in other words, I don't know what the first term in the sequence is. And because I don't even have two consecutive terms. A four and a six, 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 four is equal to a one plus DN minus one. And if we're using N equals four here, then this is just gonna be three, A six is just gonna be a one plus D. And if we're using N minus one and N A six, this is gonna be a five. Now we actually know what these numbers equal. This is equal to negative two and this is equal to six. 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 one and I need to know D, so if you notice here, what I've ended up with is I've ended up with a situation where I have two equations or I've got these two equations in which these two variables, a one and D are unknown variables. I've got two equations with two unknown variables. The whole idea here is that we're gonna turn this into a system of equations. So we've got that A one plus D and I'm gonna just simplify this. This is gonna be 3d is equal to negative two. And this is gonna be a one plus five D and that's gonna be six. That's what my two equations say. Right? I just rewrote these two 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 gonna use that here. This sort of system of equations is kind of like. So I'm gonna put is like when you had a situation where you had something like X plus three Y is equal to negative two and X plus five Y is equal to six. So remember when we had Xs and Ys and we use different methods to solve these types of problems. Well, if you already have their coefficients kind of lined up and we can use here, we could use the elimination method where we basically subtract two equations to uh sort of get rid of one of the variables and isolate one of them. And then we can use uh one equation with one variable and that was easier to solve. That's exactly what we're gonna do here. We're gonna take these two equations here and we're gonna subtract them. So remember this is the elimination method. So this is E limb. All right. So what I'm gonna do here is we're basically gonna subtract out the first terms, the A one terms over here because those coefficients are the same for equal but opposite. All right. And so when you subtract these two equations, what do you get? Well, remember you got to sort of subtract everything vertically straight down, right? So you subtract all these terms 3d minus five D gives me negative two D and then negative two minus six gives me negative eight. So if you divide by negative two on both sides, which you'll end up with is that D is equal to four. So without knowing the general formula, without even knowing the sequence, just knowing two of the numbers, I was able to figure out the common difference is equal to four. And if you think about this, this actually kind of does make sense because if a four is equal to negative two, then that means that a five, the next term in the sequence would be two. And now this totally makes sense because from two to negative sorry, negative two to positive two, that's a difference of four from 4 to 6, that's another difference of four. All right. So the common difference in the sequence is definitely four. 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 one in this, in this uh my general formula. So we've got D and we need to figure out a one, 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 gonna go ahead and pick the one with the slightly smaller numbers. So I know that a one plus 3d over here is gonna equal negative two. All right. So what I'm gonna do here is I'm just basically gonna replace D with four. So I can figure out a one. So it says here is that a one plus three times four is gonna equal negative two. So this is gonna be a one plus 12 equals negative two and I could subtract 12 from both sides. What I'll see here is that a one is equal to negative 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 all right. 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 gonna be my general formula. And what this says is that the nth term is gonna be the first term minus 14 plus D. In which case, this is four times N minus one. So this is the general formula over here. I'm gonna 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 gonna use A N uh over here is gonna be negative 14 plus four times and this is gonna be 18 minus one. So this is gonna be negative 14. So this is gonna be negative 14 plus four times 17. So if you do negative 14 times uh oh sorry plus four times 17, then what you're gonna 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. All right. So that is the 18th term in the sequence. That's how you take these types of problems where you know two 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. Uh And if not, that's ok. It's a little bit tricky. Um Let's go ahead and, and move on.