Sequences - Online Tutor, Practice Problems & Exam Prep

Sequences are ordered lists of numbers, where each number is termed a "term." They can be finite or infinite and often follow specific patterns, such as arithmetic or geometric progressions. Recursive formulas define terms based on previous terms, while general formulas express terms directly in terms of their index. Understanding these concepts is crucial for identifying patterns and deriving formulas, enhancing problem-solving skills in mathematics.

Introduction to Sequences

If you've ever had to memorize a bank account number or a phone number or even a password, you've probably had to memorize lots of numbers. But it's not enough to just know the numbers; you also have to know the order in which they come in. Well, in this video, I'm going to show you that this is exactly what a sequence is. A sequence is really just a list of numbers in a particular or specific order. So, for example, like 2, 4, 6, 8, and so on. This is a sequence. And we're going to be talking about sequences a lot in the next couple of videos. So I'm going to give you a brief introduction, and we're going to see how they're very similar to functions. We can write equations for them, and then we'll do some examples with them. So let's go ahead and get started.

The numbers in a sequence, like, for example, the 2, 4, 6, 8, are called the terms. These may also be referred to as elements or members of a sequence. And we describe them, we will label them as, you know, for example, the first term or the second term or the third term or fourth term, so on and so forth. A lot of times, what's going to happen is that your sequences will have patterns that are happening between the numbers. So, for example, 2, 4, 6, 8. It's pretty clear to see the pattern here. Each one of these numbers just increases by 2 each time. So if I wanted to figure out the 5th term in the sequence, because it's something you're very commonly going to be asked to do with these types of problems, then all we're going to have to do is just sort of add 2 to 8, and we'll get the 5th term, which is 10. Alright? Now sequences could be finite, which means that they stop after a certain value or a certain number, or they could be infinite, which means they go on forever. Most of the time, your sequences will be infinite, but sometimes you might be dealing with sequences with finite sequences. Let's just jump into our first example and see how all this works.

So here we have these 2 sequences for examples a and b, and in each one of them, we're going to find the 5th term and we're going to identify whether it's a finite or infinite sequence. Let's take a look at the first example. I've got the numbers 3, 6, 9, 12, and then some number I'm supposed to figure out, an 18. So notice how I'm supposed to figure out what the 5th term in the sequence is. Let's go ahead and look at the pattern. Just like the 2, 4, 6, 8, 10, these numbers have a pattern between them. Each one of these things increases by the same amount each time. 3, plus 3 is 6, plus 3 is 9, plus 3 is 12, so on and so forth. So, if I wanted to figure out the 5th term in the sequence, I just add 3, and I'm going to get something like 15. So that's exactly what the 5th term in the sequence is going to be. Is this finite or infinite? Well, take a look at the last term, the 18. This kind of abruptly stops, and there's nothing that goes after it, so this is a finite sequence.

Now let's take a look at the second example. We've got 1, 1 half, 1 third, 1 fourth. Many sequences will commonly include fractions, so that's perfectly fine. So what's going on here? What's the 5th number going to be? Well, if you look here, what's going on, there's a pattern between the numbers where the numerators are always 1. So no matter what happens, I'm always going to have 1 in my numerator. What's going on in the denominator? Well, if you take a look at the first term of the 1 and you rewrite it as a fraction, like 1 over 1, we can clearly see that there's a pattern with the denominators. Each one of these denominators increases by 1 each time. So the 5th term is going to be 1 over 5. You just continue on that pattern. So that's the 5th number in the sequence. Now, unlike example a, which stopped at 18, we actually have this little after the last term in the sequence, and that means that this sequence just continues going on forever. It's infinite. So whenever you see this little dot, dot, dot, that means it's an infinite sequence. Alright?

So we can see here that sequences are really just patterns of numbers. And in that regard, they're actually kind of a lot like functions. They follow specific rules, and that means that we can write equations for them. So I want to show you the difference between functions and sequences. Let's go ahead and take a look. So with a function, something like, for example, when I had, f(x)=2x, I had inputs. And when I plugged in numbers for x, which could be anything like negative numbers or decimals or even fractions and radicals, I would plug that into this formula, and I would get the outputs. I would get the f(x) or the y terms. Sequences are a little bit different because the inputs, the things that you plug into the equations, are called indexes, and we represent those letters by the letter n. So, for example, 1, 2, 3, 4, 5 are the indexes. Now these are different because these are always going to be integers. They always start at 1 and increase by 1. So I can't plug in 1.5 into this equation. I can't figure out the 1.5th term. I can only figure out the 1st, 2nd, 3rd, 4th, 5th. It's always going to be like that. We're just starting at 1 and increasing by 1. So when I plug that into an equation, like, for example, anm=2m, my inputs are going to be the ends, and the ans, the terms of the sequence, are going to be my outputs. And we represent this by a with a little subscript notation for m. So, for example, if I plug in the inputs 1, 2, 3, 4, 5 into this equation and I multiply each one of these things by 2, I'm going to get 2, 4, 6, 8, 10, which are exactly the terms that I had in the sequence at the top of the page. And these terms here can be represented by these little a's and subscripts. This the second term. This is the 3rd term, 4th term, 5th term, so on and so forth. Alright? So, really, what we can see here is that functions and sequences are very similar. You plug in numbers to calculate stuff for them. The difference really is what kinds of numbers you can plug in for functions and sequences. For functions, you can plug in anything. And so, therefore, if you were to graph this out, you would get a line. But for sequences, I can only plug in discrete values of n. 1, 2, 3, 4, 5, and that's it. Now you're never going to have to graph anything. I'm just showing these graphs here to show you the difference between functions and sequences. So now that we have a good understanding of that, let's go ahead and take a look at our second example. We're going to find the first three terms of each sequence by using the formulas that were provided here. So in example a, we have this formula, an=n2. If you were to kind of, like, sort of, sort of think about this like a function, this is actually kind of like saying f(x)=x2, and we know how to plug in numbers for this. It works the same exact way. So if I wanted to figure out the first term in sequence, all I do is I take the an=n2 and I just replace one for n anywhere everywhere I see it inside of this equation. So for the first term, this is just going to be 1 squared, and 1 squared is just 1. So what about the second term? The second term says I'm going to take the index, which is 2, and I'm going to square that. So 2 squared is 4. And for the 3rd term, a equals or or n equals 3. What I'm going to do is I'm going to take 3, and I'm going to square that. And, therefore, this 3rd term is going to be 9. So these are the first three terms of the sequence. Let's take a look at the second example. Here we have a n equals 1 over n plus 3. So let's figure out the first term. Remember, you're just going to plug in 1, 2, 3, 4, so on and so forth every time you see n in this equation. So for the first term, you're going to have 1 over this is going to be 1 +3, so this is just going to be a 1 fourth. That's going to be your term. Again, totally fine to have fractions here. So what about the second term? This is going to be 1 over, and this is going to be 2 +3. So, in other words, this is going to be 1 5th. And we're going to have 1, for sorry. For the 3rd term, you're going to do 1 over, and this is going to be 3 +3. So, in other words, this is going to be 1 over 6. And you can continue on, but these are the first three terms of the sequence. Let's take a look at our last situation or our last example. Here, what we have is the equation an=-1tothenthpower. So sometimes you can have n's and exponents, that's perfectly fine as well. Let's take a look at the first term. The first term says I'm going to take negative one, and I'm going to raise it to the positive one power. That's the n equals one power. And negative one, just by itself, is just equal to negative one. What about the second one? Well, for the second term, this is going to be negative one raised to the second power. In other words, negative one squared. So negative one squared is just going to equal positive one. Right? So the first term is negative one. The first term is positive one. You can also have even negative numbers in terms of sequences. Alright? So what about the third one? Well, a 3 is going to be negative one to the third power. This is an odd exponent, so that means that this is just going to turn into negative one again. And we're going to see that if we continue this pattern, this thing would just keep going negative one, positive one, negative one, positive one, and so on and so forth. So these are the first three terms of the sequence. Anyway, folks, so that's the basic introduction to sequences. Let's go ahead and get some practice.

The first 4 terms of a sequence are $\left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace$. Continuing this pattern, find the $7^{\th}$ term.

• A. $8\sqrt3$
• B. $6\sqrt3$
• C. $7\sqrt3$
• D. $9\sqrt3$

Determine the first 3 terms of the sequence given by the general formula

$a_{n}=\frac{1}{n!+1}$

• A. $\left\lbrace\frac12,\frac13,\frac17\right\rbrace$
• B. $\left\lbrace\frac12,\frac13,\frac14\right\rbrace$
• C. $\left\lbrace1,2,7\right\rbrace$
• D. $\left\lbrace1,\frac12,\frac16\right\rbrace$

Writing a General Formula

Welcome back, everyone. So when we introduced sequences, we saw equations for a_n involving n. The idea was that these are your inputs. You plug in 1, 2, 3 into the equations, and then you get the terms out of it. But sometimes that's going to be backwards. You're going to have to look at a sequence of numbers or terms, and you're going to have to figure out what's the formula for it. And to do that, I'm going to show you in this video that we're going to be writing general formulas, or sometimes they're called explicit formulas. These are equations involving n where you plug in n and you get the general terms out of it. What I'm going to show you here is that it really just comes down to finding the patterns that are going on between the numbers. So I'm going to show you a bunch of different examples of the most common types of patterns because when you see them, it means that your formula is going to contain some of these expressions. We're going to work a bunch of examples together. Let's go ahead and get started.

Alright. So let's take a look at the first pattern here. So oftentimes in your sequences, you'll see numbers that increase by the same amount each time. So for example, 5, 6, 7, 8, 9. If you look at these, there's a pattern here. All the numbers increase by 1 each time. And we've also seen other examples where numbers increase by 2, like 2, 4, 6, 8, 10, or 3, like 3, 6, 9, 12. In all those cases where you have numbers increasing by 1 or 2 or 3, what that means is that your formula contains the expression n or 2n or 3n or something like that. Right? So on and so forth. So our general formula over here is going to include something like n. Now is this it? Is this the formula? Are we done? Well, you could always just plug in numbers just to sort of check and make sure that you're getting the right numbers in your sequence. The first term in the sequence is 5. So if I plug in n=1 into this expression, all I'm going to get here is 1. So what's going to happen here is that we're often going to have to adjust our formula by adding, subtracting, multiplying, and dividing numbers or constants to get the sequence that we need. What you'll notice is that all of these numbers increase by 1 just like the index does, but the problem is that the first number isn't 1, it's 5. So what we have to do is that the first number is sort of shifted up by 4, so what you're going to have to do is you're going to have to add 4 to this, this formula over here. Now if you go ahead and check the first expression by adding 4, you're going to see that the number is 5, and that's exactly what the first term is. If you plug in n=2, you'll get 2+4, which equals 6. That's exactly what the second term is, and then so on and so forth. So what you'll see here is that n by itself wasn't enough. You're going to have to adjust your formula by adding a constant here. So that's the general formula for this sequence. Alright?

Now let's move on to the second scenario here. A lot of times, you're going to see sequences that alternate signs. We've already seen an example of this with negative 1, positive 1, negative 1, positive 1. In our example here, we have negative 5, positive 5, negative 5, positive 5, so on and so forth. So we have the same number, but it flips the sign. Whenever that happens, that means you're going to have -1 raised to some power of n. Because what happens is as n goes bigger, you're going to have this exponent that oscillates between even and odd numbers, giving you even, giving you positive and negative numbers. And, really, what it comes down to is you're going to look at the first number in your sequence. And if it's negative, then you have this expression over here. If it's positive, then you have this expression over here. So in our example, the first sign that we see in this sequence is negative. So that means that our general formula is going to contain -1 raised to the n power. So is this all? Well, let's just go ahead and just sort of test some numbers here. A 1, this is going to be -1 raised to the one power. Now that's just negative one. How do I get 5 out of this? Do I add a number to this or subtract the number? Well, what happens is if you add a number like, let's say, 4, then you're going to add 4, but then on the next term, it's going to be -1+4, and that's not going to give you 5. So we can't add a number here, and, instead, what we're going to have to do is we're going to have to multiply a number. In order for the first number to be 5, I'm going to have to multiply this whole thing and adjust it by multiplying by 5. Now what you'll see here is that if you multiply this expression by 5, the first term will be -1×5, which is negative 5. The second term will be -122×5, which turns to a positive times 5, and that equals positive 5. And then the whole thing will repeat over and over again. Right? So what you'll see here is that we had to adjust this formula by multiplying by 5, and this is the formula for your sequence. Let's move on to the third situation here, which is that sometimes your sequences may contain fractions. So, in this sequence, we have 12, 23, 34, 45, 56. Whenever this happens, it means that your general formula is also going to contain fractions. And usually, what it's going to happen here is that the tops and the bottoms, the numerators and denominators are going to have slightly different patterns. So let's take a look at this, this situation over here. If you look at the numerators, what you'll see is that each one of these numbers, kind of like the first example that we did, they all increased by 1. It goes 1, 2, 3, 4, 5. And that's actually exactly what the n the index of your sequence does. Right? It also starts at 1 and it goes 1, 2, 3, 4, 5. So in this situation, when we have, things, you know, that increase by 1, it means that you're going to have an n inside of that expression. So in the numerator, this is actually just going to be n. And if you look at this, we actually won't even need to adjust it because for n=1, n=2, n=3, n=4, n=5, you'll exactly get these numerators, 1, 2, 3, 4, 5. Let's take a look at the bottoms here. This goes 2, 3, 4, 5, and 6. So just like the top, each one of these numbers also increases by 1, so so we know we're going to have to have an n here in the denominator. However, the starting number, kind of like this 5, 6, 7, 8, 9, isn't 1. It starts at 2 and then goes 3, 4, 5, 6. So we're going to have to adjust this a little bit, and we're going to have to do n+1 so that we make sure that the first number is 2. So if you look at this and try to plug in some numbers really quick, a one is going to be 11+1, which is going to give you 12. That's exactly what that first term is. If you go ahead and plug in the second number, this is going to be 22+1. And that's going to give you 23. That's exactly what the second number is, and so on and so forth. So we had to adjust this formula, and this turns out to be the general formula for this sequence over here. Alright?

So the last last but not least, sometimes you may see sequences increase exponentially. Where if you look at these numbers here, 2, 4, 8, 16, 32, this is not like 5, 6, 7, 8, 9 because the difference between these numbers constantly changes. Here, it's 2. Here, it's 4. Here, it's 8, and here, it's 16. So this is an example of an exponential sort of type of sequence where the numbers are getting bigger faster. And whenever this happens, your formula contains some number raised to the n power. So let's take a look here. So we have a_n= some number over here that's going to be raised to the n power. Alright? Now how does this work? Well, if I try to plug in something like 1, that's not going to make any sense because 1 raised to the n power is always just going to be 1. So let's take a look here. Notice that the first number is 2, and then all of these things are actually powers of 2. So, actually, we're just going to stick a 2 into this sequence here. And the general formula for this is going to be 2n. We're actually going to talk about these sequences more later on, but this is the general formula for this sequence. And if you plug in some numbers here, you'll see that it makes sense. 2n raised to the one power is 2. For the second term, a 2, this is going to be 2n raised to the 2 power that equals 4, and so on and so forth. Alright? So we get our numbers there. Alright, folks. So that's it for this one. Again, these are the sort of most common situations that you'll see. Hopefully, this made sense, and let's get some practice.

Example 1

Everyone, welcome back. So in this example problem, we're given the first four terms of a sequence. We've got these four terms separated by commas, and we're going to use this sequence of numbers to write the general formula. So in other words, we're going to look at the patterns of numbers and figure out if we can find an equation for an, the general or nth term. And that's just going to be some equation, and then we're going to use this equation that we find to find the 15th term of the sequence. So rather than having to continue out the pattern out to the 15th term, we're going to be able to calculate this very quickly just by plugging 15 in for n once we figure out our formula. Alright? So let's go ahead and get started here.

If you look at this, remember that when we look at the sequence of numbers, we're going to look at the patterns of numbers to figure out what's going to be in our formula. So there's a couple of things. For example, if you notice that numbers are increasing by the same amounts for each term or if you see alternating signs, things like that. Let's take a look at our sequence. We have 1 over 1 times 2, 1 over 2 times 3, 1 over 3 times 4, 1 over 4 times 5. So this is a fraction. And remember, whenever we have fractions, that means that your general formula is going to have a fraction. So that's a good place to start.

And now we're going to take a look at the top and the bottom because those patterns will usually be different. For example, notice how the numerator, the tops of each one of the fractions, no matter what the term is, is always 1. What that tells us is that this top doesn't depend on what the index is. It has nothing to do with n. So whenever you see ones on the top of each one of your numbers, that just means that there's going to be a one in your general formula because this thing, no matter what n is, is always just going to be 1. Alright? So that one's pretty easy, the numerator.

Let's take a look at the denominator because it's a little bit different. Right? You can notice here that we're going to actually have two numbers that are always being multiplied in the denominator. 1 times 2, 2 times 3, 3 times 4, 4 times 5. And also, notice how the numbers are always different each time. In other words, we here we have 1 times 2, 2 times 3. Right? So the numbers are constantly changing across the denominator. So what that tells us is that we're definitely going to have some kind of an n in our denominator.

Let's take a look at the first term in each of the denominators because this one starts at 1 and then the next, in the next term, the first number is a 2. In the next term, the first number is a 3. In the next term, the first number is a 4. So in other words, between each one of the first terms of the denominator, we're increasing by the same amount each time. This is actually like plus 1. Now remember, whenever we have numbers that increase by the same amount each time, then that's going to be some kind of a multiple of n. And when it increases by 1, that's just going to be n. And notice how over here, what happens is that the first term, the index of 1, corresponds with the first number being 1. An index of 2, the second term means the second number is a 2. The 3rd term, that first number is a 3. And the 4th number, that first number is a 4. So that means that this is actually literally just n. Right? Because as the index changes as we go from 1, 2, 3, 4, then this number is going to be 1234. Alright? That first number is definitely going to be n in our general formula.

Now let's take a look at the second number over here. The second number in the denominator is 2 and then 3 and then 4 and then 5. So what you'll notice here is that between the terms, the second number in the denominator also increases by 1. So does that just mean that we're just going to have to multiply n times n? Well, if you think about it, not exactly because 1n×n means that if you were to plug in an index of 1, this would be 1 over 1 times 1. And that's not what the first term tells us. It's 1 over 1 times 2. So notice how it happens is that this index over here of 1, this number is always one higher than that index. Right? So here for the second term, we have the number 3. For the 3rd term, the index of 3, that second number is a 4. So it's definitely going to be n, but remember that sometimes you're going to have to adjust your formula by adding, subtracting, or multiplying, and dividing constants. And in this case, because everything is shifted up by 1, then in our formula, we have to include n plus 1.

So now what happens is if you plug in n equals 1, you're going to get 1n×n+1, which is going to be 1 over which is going to be 2. So that's how we adjust for that number, kind of starting at some number that isn't 1. So we're going to have to raise this by 1. So if you plug in, so, for example, just n equals 1, which you're going to see, this is going to be 1 over. Now this is the index of 1, so we're going to plug in 1 times, and then this is going to be 1 times 1 plus 1. So this is going to be 2. If you plug in an index of 2, remember, this is just going to the first term is going to be 2, and then the second term is going to be 2+1. So this is 2+1, and this is going to be 1 over 2 times 3. So notice how if you keep sort of, if you keep actually extending out the numbers, you'll see that the a 1, a 2, a 3, a 4 are exactly match what's going on in the sequence. Alright? So this definitely is the general formula for our sequence over here. It's 1 over n times n plus 1. Sometimes you might see this in parentheses or something like this if there's if there's, you know, multiplication that's going on. So that's really what our general formula i

Example 2

everyone. Welcome back. So in this example, we're going to do something very similar where you have the first five numbers of a sequence, negative 2, 4, negative 6, 8, negative 10, and so on and so forth. And in this problem, we're asked to again write the general formula for the nth term, and we're going to use this to calculate the 18th term. So, again, these formulas are really useful for calculating really, really high terms because rather than having to sort of follow-up the pattern for 18 terms, we're going to be able to figure out a formula for \( a_n \). Right? That's just going to be some formula over here. And then we're going to use this to figure out the 18th term by basically just plugging in 18 for \( n \). Alright? So let's go ahead and get started here. Remember, there are lots of things to consider when writing general formulas. Let's look at the pattern of numbers. So do we see fractions? We don't see any fractions. Do we see numbers that increase by the same number each time? Well, if you take a look here, what happens is that from negative 2 to 4, that's an increase of 2. I'm just looking at the number, not the sign. From 4 to 6, I also have an increase of 2, 6 to 8, increase of 2, and increase of 2. So I've got the same number that sort of it increases each time, but now I've got this other thing that's happening as well, which is that the signs alternate. I've got negative and then so I've got negative here then positive, negative here then positive, negative, and then so on. So, actually, I'm going to sort of combine some of the rules that I've seen before. So remember, whenever you have alternating signs, that always means that there's going to be a negative one to the nth power. And remember if the first term is is is, negative, that means that we're going to have an \( n \) and not \( n+1 \). So if the first term is negative, then it's just going to be negative one to the nth power. Now, obviously, this isn't just enough by itself because if I do negative one to the nth power, that's just going to alternate from negative one to positive one. So we're going to have to figure out how to increase these numbers by 2 each time. So what do I do? Remember, a lot of times you're going to have to multiply constants or different things like that. So can I just multiply this whole sequence by 2? Well, what happens here is if I just carry out this sequence and figure out the couple of numbers, I'm going to have negative 1, positive 1, negative 1, positive 1 times 2. So in other words, it's just going to be negative 2, positive 2, negative 2, positive 2. So that's not enough either. Now remember, whenever you increase by the same number each time, like plus 1 or plus 2 or plus 3, that means that you're going to have either \( n \) or \( 2n \) or \( 3n \) or so on. So in this case, what's going to happen is we're going to have to increase or going to have to multiply this by \( 2n \). Alright? So this \( 2n \) is now going to account for the fact that now for each term, you're going to increase by 2 as the index gets higher. So now let's take a look at the first couple of terms here and see if this sort of lines up with the terms that we have. So for \( a_1 \), this is going to be negative one to the one power times two times one. So in other words, this is just going to be negative one times 2, which is negative 2. That's exactly what the first term is. Let's take a look at the second term. This is going to be negative one to the second power times 2 times 2. So in other words, this is just going to be positive one times 4. That's going to give us positive 4. That's exactly what the second term is. One more just for, you know, just to be safe. This is going to be negative one to the third power times two times three. So in other words, this is going to be negative 1, right, because it's to the 3rd power. It's an odd power. And this is going to be times 6, so this is going to be negative 6. And that's exactly where our 3rd term is. So it turns out that this is the general formula for the sequence here, negative one to the nth power and then times \( 2n \). Now what's also really important about this is that you can't just sort of merge these two things in. So, for example, you can't say something like negative \( 2n \) to the nth power. It doesn't work like that. You can't like absorb something into that parentheses. Alright? So you're just going to have to leave it like this general formula. So now how would we figure out the 18th term? Very simple. We just plug in 18 for the \( n \) for the index in this formula. So the 18th term is just going to be negative one raised to the 18th power times 2 times 18. Now if you plug this into your calculator, what you should see is that negative one to the 18th power is just 1, right, because it's an even power, and then 1 times 36 just equals positive 36. So that's the general formula. And then for the 18th term, we just have positive 36. Alright? So thanks for watching. Hopefully, this made sense, and I'll see you in the next one.

Recursive Formulas

Everyone, in previous videos, we saw how to use the general formula to figure out the terms of a sequence. For example, if I had something like \( a_n = 2n \), then the whole idea was I can grab these indexes, 1, 2, 3, 4, 5. I plug them into this equation here to get my outputs. What we saw is that the first five terms of the sequence were 2, 4, 6, 8, 10. What I'm going to show you in this video is that sometimes you may be asked to use or write a different kind of formula called a recursive formula. I'm going to show you the difference between general and recursive formulas, and, basically, what it is is that recursive formulas tell you how to find terms, your \( a_n \) terms, but not based on \( n \) that you plug into the equation, but it's actually based on the previous terms, the terms that go before in the sequence. I'm going to break down the difference, and we'll do some examples together. Let's get started.

So like general formulas, recursive formulas also tell you how to calculate terms of a sequence. We had something like \( a_n = 2n \). You plug these values in and you get your numbers that way. But now let's look at a different type of formula. This formula says \( a_n \), which is the next term in the sequence, is equal to \( a_{n-1} + 2 \). What this really what this \( a_{n-1} \) is, is it's basically just the previous term in the sequence. It's the index \( n \) subtracted by 1, so it's the previous term. So really, let's take a look here. Let's say you have this formula, \( a_{n-1} + 2 \), and you have the first term in the sequence, which is \( a_1 \). Let's calculate the next few terms of the sequence using this formula.

Alright. So if I wanted to calculate \( a_2 \), what this formula tells me is I'm going to have to use the previous term in the sequence, \( a_1 \), and I'm just going to have to add 2 to it. Alright? And so what this \( a_1 \) is, it's just 2, so \( 2+2 = 4 \). Now let's calculate the next term, this \( a_3 \). What this formula says is that to calculate \( a_3 \), I need \( a_2 \), and I have to add 2 to it. We actually just calculated what 2 is, that's just the 4 that we just calculated. So, I'm going to have to take 4, add 2 to it, and now I get 6. So, now I have 4 and 6. If I wanted to calculate \( a_4 \), you'll see the pattern here: I have to know what \( a_3 \) is, and so on and so forth. So, you're just going to get 6 plus 2, and you're just going to get 8. If we continue on this pattern, what you're going to see is that we get 4, 6, 8, 10. Notice how we have actually ended up with the same exact numbers that we did when we did this with the general formula, but we just use a totally different formula to get there. So the basic difference between general and recursive formulas is that for the general formula, you're going to need \( n \) to plug into this equation to get the nth term. So you need to know what \( n \) is. And for a recursive formula, you just need the previous term in the sequence, this \( a_{n-1} \), to get what the next term is. That's the main difference.

Now, I want to point out that you might be thinking, well, isn't the general formula always going to be better? And not necessarily. Sometimes you may be asked to just find the next few terms of a sequence, and finding the general formula, if you're not given it, might be really hard. So it's just easier to sort of tell what the pattern is between these numbers. Hey. Look at these. All these are just increasing by 2, so I'm just going to continue on that pattern. So it's not that one is always better than the other. It just depends on what you're given to ask. Let's go ahead and take a look at some examples here and work this one out together.

Alright? So given this recursive formula and the first term of each sequence, I want to find the next three terms in the sequence. Alright? So I'm told here that in example \( a \), \( a_n = 2 \times a_{n-1} + 3 \), that's the previous term, and then I'm going to have to add 3 to it. And I'm already told what \( a_1 \) is, that's just equal to 1. Let's use this formula to now find the 2nd term. The 2nd term says, or this formula says, in order to find \( a_2 \), I'm going to have to take 2, multiply it by the previous number in the sequence, which is \( a_1 \), and then I'm going to have to add 3. But notice that we've already calculated that \( a_1 \) is 1. This is going to be \( 2 \times 1 + 3 \). If you work this out, what you're going to get is 5. So the second term of the sequence is 5. Now let's take a look at the third one. This \( a_3 \) says, in order to calculate the next term, I'm going to have to take 2 and multiply it by the previous term \( a_2 \) and add 3. We've already figured out what \( a_2 \) is. It's just 5 over here. So this is just going to be \( 2 \times 5 + 3 \), and that gives me 13. Now for the 4th one, this is going to be \( 2 \times the previous number a_3 + 3\). You've already figured that out. That's just 13. So this is going to be \( 2 \times 13 + 3 \), and this is going to equal 29. Alright? So those are the first four terms in the sequence. Notice how, if I just look at these numbers and try to come up with a general formula, it might be really tricky to do this. So if you're just asked to figure out the next few terms of a sequence, usually, the recursive formula is going to be really, really good for this.

Alright? So here are your answers. Let's take a look now at the second one. In fact, actually, pause the video and see if you can try it on your own. So in the second formula here, what I've got is I've got \( a_n = n \times the previous term in the sequence \). And I've got \( a_1 \), which equals 1. What's the second term? Well, \( a_2 \) just tells me that it's just going to be the index itself, in which case I'm using \( n = 2 \). So this is going to be \( 2 \times a_1 \). So in other words, this is just going to be \( 2 \times the previous number, which is 1, and that gives me 2 \). So that is \( a_2 \). So now let's calculate \( a_3 \). This is where \( n = 3 \). Right? So what I'm going to do is this formula says take the index, which is 3 in this case, and now multiply it by the previous number, which is \( a_2 \). So what is that? So this is going to be \( 3 \times 2 \), and that gives me 6. That's the 3rd term in the sequence. So now for \( a_4 \), hopefully, you see the pattern now. This is where \( n = 4 \). You're going to grab the index, which is 4, multiplied by the previous term, which is \( a_3 \). This is going to be \( 4 \times 6 \), and this is going to be 24. Alright? That's going to be 24 over here. Alright? And that's how you use these, these recursive formulas to figure out the next few terms of a sequence. Hopefully, that made sense. Thanks for watching, and let's get some practice.

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.

• A. $\left\lbrace1,1,2,3,5,18\right\rbrace$
• B. $\left\lbrace1,2,3,5,8,13\right\rbrace$
• C. $\left\lbrace0,1,1,2,3,5\right\rbrace$
• D. $\left\lbrace1,1,2,3,5,8\right\rbrace$

Here’s what students ask on this topic:

What is a sequence in mathematics?

A sequence in mathematics is an ordered list of numbers, where each number is called a term. The terms are arranged in a specific order, and the sequence can be either finite (having a limited number of terms) or infinite (continuing indefinitely). Sequences often follow specific patterns, such as arithmetic or geometric progressions. For example, the sequence 2, 4, 6, 8, ... is an arithmetic sequence where each term increases by 2. Understanding sequences is crucial for identifying patterns and deriving formulas in mathematics.

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

To find the nth term of an arithmetic sequence, you can use the formula:

$a_n=a_1+(n-1)d$

where $a_{\mathrm{n}}$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. For example, in the sequence 3, 6, 9, 12, ..., the first term $a_1$ is 3, and the common difference $d$ is 3. To find the 5th term, plug in the values: $a_5=3+(5-1)3=15$.

What is the difference between a general formula and a recursive formula in sequences?

A general formula, also known as an explicit formula, expresses the nth term of a sequence directly in terms of n. For example, in the sequence defined by $a_n=2n$, you can find any term by plugging in the value of n. A recursive formula, on the other hand, defines each term based on the previous term(s). For example, in the sequence defined by $a_n=a_{n-1}+2$ with $a_1=2$, you need to know the previous term to find the next term. General formulas are useful for finding any term directly, while recursive formulas are useful for understanding the relationship between consecutive terms.

How do you determine if a sequence is finite or infinite?

A sequence is finite if it has a limited number of terms, meaning it eventually stops. For example, the sequence 1, 2, 3, 4, 5 is finite because it ends at 5. A sequence is infinite if it continues indefinitely without stopping. For example, the sequence 1, 2, 3, 4, ... is infinite because it goes on forever. You can often identify an infinite sequence by the presence of ellipses ("...") indicating that the sequence continues. Understanding whether a sequence is finite or infinite is important for determining the scope and behavior of the sequence.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$. For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3. The general formula for the nth term of an arithmetic sequence is:

$a_n=a_1+(n-1)d$

where $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. Arithmetic sequences are useful in various mathematical applications, including solving problems involving linear patterns.