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 gonna show you that this is exactly what a sequence is. A sequence is really just a list of numbers in a part in a particular or specific order. So for example, like 2468 and so on. So this is a sequence and we're gonna be talking about sequences a lot in the next couple of videos. So I'm gonna give you a brief introduction and we're gonna see how they're very similar to functions. We can write equations for them and we'll do some examples with them. So let's go ahead and get started. So the numbers in a sequence like for example, the 2468 are called the terms, these are also may 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. Now, a lot of times what's gonna happen is that your sequences will have patterns that are happening between the numbers. So for example, 2468, it's pretty clear to see the pattern here. Each one of these numbers just increases by two, each time. So if I wanted to figure out the fifth term in the sequence, because it's something you're very commonly gonna be asked to do with these types of problems. Then all we're gonna have to do is just sort of add 2 to 8 and we'll get the fifth term which is 10. All right. Now, sequences could be finite, which means that they stop after a certain value or a certain number or they can be infinite, which means they go on forever. Most of the time your sequences will be infinite. But sometimes you may be dealing with s with finite sequences. Let's just jump into our first example and see how all this works. So here we have these two sequences for examples A and B and in each one of them, we're gonna find the fifth term and we're gonna identify whether it's a finite or infinite sequence. Let's take a look at the first example, I've got the numbers 369, 12 and then some number I'm supposed to figure out in 18. So notice how I'm supposed to figure out what the fifth term in the sequence is let's go ahead and look at the pattern just like the 2468, 10. These numbers have a pattern between them. Each one of these things increases by the same amount. Each time. Three plus three is six plus three, is nine plus three, is 12. So on and so forth. So if I wanted to figure out the term in the sequence, I just add three and I'm gonna get something like 15. So that's exactly what the fifth term in the sequence is gonna be. Is this finite or infinite? Well, take a look at the uh 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 11 half, one third, 1/4 many sequences will commonly include fractions that's perfectly fine. So what's going on here? What's the fifth number gonna be? Well, if you look here, what's going on, there's a pattern between the numbers where the numerators are always one. So no matter what happens, I'm always gonna have one in my, in my numerator, what's going on in the, in the denominator? Well, if you take a look at the first term of the one and you rewrite as a fraction like 1/1, we can clearly see that there's a pattern with the denominators. Each one of these denominators increases by one each time. So the fifth term is gonna be 1/5, you just continue on that pattern. So that's the fifth number in the sequence. Now, unlike this example, a which stopped at 18, we actually have this little dot dot dot 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. All right. 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 wanna 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 uh FX is equal to two X, 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 output. So I would get the FX or the Y terms sequences are a little bit different because the inputs, the things that you plug into the equations are called index and we represent those letters by the letter N. So for example, 12345 are the indexes. Now, these are different because these are always gonna be integers, they always start at one and increase by one. So I can't plug in 1.5 into this equation. I can't figure out the one point fifth term. I can only figure out the 1st, 2nd, 3rd, 4th, 5th, it's always gonna be like that just starting at one and increasing by one. So when I plug that into an equation, like for example, A N equals to M, my inputs are gonna be the ends and the A ns, the terms of the sequence are gonna 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 12345 into this equation and I multiply each one of these things by two, I'm gonna get 2468 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 uh A's and subscripts. This is the second term, this is the third term, fourth term, fifth term, so on and so forth. All right. 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 12345. And that's it. Now, you're never gonna have to graph anything like just showing these graphs here to show you the difference between functions and sequences. So now we have a good understanding of that. Let's go ahead and take a look at our second example. We're gonna find the first three terms of each sequence by using the formulas that were provided here. So in example, a, we have this formula A and equals N squared if you were to kind of like sort of uh sort of think about this like a function. This is actually kind of like saying FX is equal to X squared 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 the sequence, all I do is I take the A N equals N squared and I just replace one for N anywhere everywhere I see it inside of this equation. So for the first term, this is just gonna be one squared and one squared is just one. So what about the second term? The second term says I'm gonna take the index which is two and I'm gonna square that. So two squared is four and for the third term A equals or, or N equals three what I'm gonna do is I'm gonna take three and I'm gonna square that. And therefore this third term is going to be nine. 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 one over N plus three. So let's figure out the first term. Remember you're just gonna plug in 1234, so on and so forth every time you see N in this equation. So for the first term you're gonna have one over, this is gonna be one plus three. So this is just gonna be a 1/4 that's gonna be your term. And again, totally fine to have fractions here. So what about the second term? This is gonna be one over and this is gonna be two plus three. In other words, this is gonna be 1/5 and we're gonna have one sorry for the third term you're gonna do one over and this is gonna be three plus three. So in other words, this is gonna be 1/6 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 A N equals negative one to the nth power. So sometimes you can have NS and exponents that's perfectly fine as well. Let's take a look at the first term, the first term says I'm gonna take negative one and I'm gonna 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? For the second term? This is gonna be negative one raised to the second power. In other words, negative one squared. So negative one squared is just an equal positive one, right? So the first term is negative one, the first term is positive one. So you can also have even negative numbers in terms of sequences. All right. So what about the third one? Well, a three is gonna be negative one to the third power. This is an odd exponent. So that means that this is just gonna turn into negative one again. And we're gonna 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.

2

Problem

Problem

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

$9\sqrt3$

3

Problem

Problem

Determine the first 3 terms of the sequence given by the 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 123 into the equations and then you get the terms out of it. But sometimes that's gonna be backwards, you're gonna have to look at a sequence of numbers or terms and you're gonna have to figure out what's the formula for it. And to do that, I'm gonna show you this video that we're gonna be writing general formulas. Sometimes it's called explicit formulas. These are equations involving N where you plug in NS and you get the general terms out of it. So what I'm gonna show you here is that it really just comes down to finding the patterns that are going on between the numbers. So I'm gonna 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 gonna contain some of these expressions. We're gonna work a bunch of examples together. Let's go ahead and get started. All right. 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, 56789, if you look at these, there's a pattern here, all the numbers increase by one each time. And we've also seen other examples where numbers increase by two, like 2468, 10 or three, like 369, 12 in all those cases where you have numbers increasing by one or two or three. What that means is that your formula contains the expression N or two N or three N or something like that, right? So on and so forth. So our general formula over here is gonna include something like N. Now is this, it, is this the formula are we done or 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 five. So if I plug in N uh equals one into this expression, all I'm gonna get here is one. So what's gonna happen here is that we're oftentimes gonna 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 one just like the index does. But the problem is that the first number isn't one, it's five. So we have to do is that the first number is sort of shifted up by four. So what you're gonna have to do is you're gonna have to add four to this, uh, this formula over here. Now, if you go ahead and check the first expression by adding four, you're gonna see that the number is five. And that's exactly what the first term is. If you plug in N equals two, you'll get two plus four, which equals six. 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 gonna have to adjust your formula by adding a constant here. So that's the general formula for this sequence. All right. Now, let's move on to the second scenario here. A lot of times you're gonna see sequences that alternate signs. We've already seen an example of this with negative one, positive, one, negative, one, positive one. In our here, we have negative five, positive five, negative five, positive five so on and so forth. So we have the same number but it flips the sign, whatever that happens, it means you're gonna have negative one raised to some power of N because what happens is as N goes bigger, you're gonna have this exponent that oscillates between even and odd numbers giving you even uh giving you positive and negative numbers. And really what it comes down to is you're gonna 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 gonna contain negative one raised to the end power. So is this all? Well, let's just go ahead and sort of test some numbers here. A one, this is gonna be negative one raised to the one power. Now that's just negative one. How do I get five 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 four, then you're gonna add four. But then on the next term, it's gonna be negative one plus four and that's not gonna give you five. So we can't add a number here. And instead what we're gonna have to do is we're gonna have to multiply a number in order for the first number to be five. I'm gonna have to multiply this whole thing and adjust it by multiplying by five. Now, what you'll see here is that if you multiply this expression by five, the first term will be negative one times five, which is negative five, the second term will be negative one squared which turns to a positive times five and that equals positive five. 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 five. And this is the formula for your sequence. Let's move on to the third situation here, which is that sometimes your sequences may, may contain fractions. So in this sequence, we have 1/2 2/3 3/4 4/5 5/6. And whenever this happens, it means that your general formula is also gonna contain fractions. And usually what it's gonna happen here is that the tops and the bottoms, the numerations denominators are gonna have slightly different patterns. So let's take a look at this, uh this situation over here, if you look at the numerators, which you'll see is that each one of these numbers kind of like the first example that we did, they all increase by one. It goes 12345. And that's actually exactly what the N, the index of your sequence does, right? It also starts at one and it goes 12345. So in this situation, when we have uh things you know that increase by one, it means that you're gonna have an N inside of that expression. So in the numerator, this is actually just gonna be N and if you look at this, we actually won't even need to adjust it because for N equals 12345, you'll exactly get these numerators 12345. Let's take a look at the bottoms here, this goes 2345 and six. So just like the top, each one of these numbers also increases by one. So we know we're gonna have to have an end here in the denominator. However, the starting number kind of like this 56789 isn't one, it starts at two and then goes 3456. So we're gonna have to adjust this a little bit and we're gonna have to do N plus one so that we make sure that the first number is two. So if you look at this and try to plug in some numbers really quick. A one is gonna be one divided by uh one plus one, which is gonna give you one half. That's exactly what that first term is. If you go ahead and plug in the second number, this is gonna be 22, divided by two plus three and this is gonna, I'm sorry, two plus one and that's gonna give you two thirds. 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. All right. So the last last but not least, sometimes you may see sequences increase exponentially where if you look at these numbers here, 248, 1632. This is not like 56789 because the difference between these numbers constantly changes. Here. It's two. Here, it's four. Here, it's eight and here it's 16. So this is an example of an exponential sort of type of sequence where uh 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 equals some number over here that's gonna be raised to the N power. All right. Now, how does this work? Well, if I try to plug in something like one, that's not gonna make any sense because one raised to the N power is always just gonna be one. So let's take a look here. Notice that the first number is two and then all of these things are actually powers of two. So actually, we're just gonna stick a two into this sequence here. And the general formula for this is gonna be two to the end power. We're actually gonna talk about these sequences more later on. Uh But this is the general formula for this sequence and if you plug in some numbers here, you'll see that it makes sense to race to the one power is two. For the second term A two, this is gonna be two, race to the two power that equals four and so on and so forth. All right. So we get our numbers there. All right 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.

5

example

Example 1

Video duration:

5m

Play a video:

Hey, 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 gonna use this sequence of numbers to write the general formula. So in other words, we're gonna look at the patterns of numbers and figure out if we can find an equation for a N, the general or nth term. And that's just gonna be some equation. And then we're gonna use this equation that we find to find the 15th term of the sequence. So rather having the having to continue out the pattern out to the 15th term, we're gonna be able to calculate this very quickly just by plugging 15 in for N once we figure out our formula. All right. So let's go ahead and get started here. So if you look at this, remember that when, when we look at the sequence of numbers, we're gonna look at the patterns of numbers to figure out what's gonna be in our formula. So there's a couple things. So 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 1/1 times 2, 1/2 times 3, 1/3 times 4, 1/4 times five. So this is a fraction. And remember whenever we have fractions, that means that your general formula is gonna have a fraction. So that's a good place to start. And now we're gonna take a look at the top and the bottom because those patterns will usually be different. For example, notice how the num numerator, the tops of each one of the fractions, no matter what the term is, is always one. But that tells us is that this top doesn't depend on what the index is and has nothing to do with M. So whenever you see ones on the top of each one of your numbers, that just means that there's gonna be a one in your, in your uh general formula because this thing no matter what N is, is always just gonna be one. All right. So that one's pretty easy. The numerator, let's look at the denominator because it's a little bit different, right? You can notice here that we're gonna actually have two numbers that are always being multiplied in the denominator one times 22 times 33 times 44 times five. And also notice how the numbers are always different each time. In other words, we hear we have one times two times three, right? So the numbers are constantly changing across the denominator. So what that tells us is that we're definitely gonna have some kind of an N in our denominator. So let's take a look at the first term in each of the denominators because this one starts at one and then the next, uh, in the next term, the first number is a two in the next term. The first number is a three, the next term, the first number is a four. 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 one. Now remember whenever we have numbers that increase by the same amount each time and that's gonna be some kind of a multiple of N and when it increases by one, that's just gonna be N. And notice how over here what happens is that the first term, the index of one corresponds with the first number being one, an index of two. The second term means that the sec the that number is a two, that the third term, that first number is a three and the fourth number, that first number is a four. So that means that this is actually literally just N, right? Because as the index changes as we go from 1234, and this number is gonna be 1234. All right, that first number is definitely gonna be N in our general formula. Now, let's take a look at the second number over here. The second number in the, in the denominator is two and then three and then four and then five. So what you'll notice here is that between the terms, the second number in the denominator also increases by one. So does that just mean that we're just gonna have to multiply N times N? Well, if you think about it, not exactly because one over N times N means that if you were to plug in an index of one, this would be 1/1 times one. And that's not what the first term tells us. It's 1/1 times two. So notice how it happens is that this index over here of one, this number is always one higher than that index, right? So here for the second term, we have the number three for the third term, the index of three, that second number is a four. So it's definitely gonna be N but remember that sometimes you're gonna have to adjust your formula by adding subtracting or multiplying and dividing constants. And in this case, because everything is shifted up by one. And in our formula, we have to include N plus one. So now what happens is if you plug in N equals one, you're gonna get 1/1 times one plus one, which is gonna be one over. It was using the two. So that's how we adjust for that number kind of starting at some number that isn't one. So we're gonna have to raise this by one. So if you plug in, so for example, just N equals one, which you're gonna see, this is gonna be one over. Now this is the index of one. So we're gonna plug in one times and then this is gonna be one times one plus one. So this is gonna be two. If you plug in an index of two, remember this is just gonna be the first term is gonna be two and then the second term is gonna be two plus one. So this is two plus one and this is gonna be 1/2 times three. So notice how if you keep sort of a, you know, if you keep actually extending out the numbers, you'll see that the A one, a two, a three, a four are exactly match what's going on in the sequence. All right. So this definitely is the general formula for our sequence over here. It's one over N times N plus one. Sometimes you might see this in parentheses or something like this if there's, if there's, you know, multiplication that's going on. Um So that's really what our general formula is. So how do we use this now to find the 15th number in the sequence? Well, it's pretty straightforward basically what the, what this A N formula tells us is that to find out the 15th term, we're just gonna do 15 as our end. So in other words, this is gonna be 15 times and this is gonna be 15 plus one. All right. So it's just gonna be the next number up. So this is just gonna be 1/15 times 16. And if you actually multiply this out in your calculators, which you should see is that this is about 0.0042 repeating. Uh It's gonna be 416, but it's just gonna round 20042. All right. So that is the 15th term in your sequence and that's the general formula. So thanks for watching and I'll see you in the next one.

6

example

Example 2

Video duration:

4m

Play a video:

Hey, everyone, welcome back. So in this example, we're to do something very similar where you have the first five numbers of a sequence negative 24, negative 68, 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 gonna 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 out the pattern for 18 terms, they're gonna be able to figure out a formula or A N, right. That's just gonna be some formula over here. And then we're gonna use this to figure out the 18th term by basically just plugging in 18 for N. All right. So let's go ahead and get started here. Remember there's lots of things to consider when writing general formulas, but 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 two. I'm just looking at the number, not the sign from 4 to 6. I also have an increase of 26 to 8, increase of two, an increase of two. So I've got the same number that sort of it, it increases each time. But now I've got this other thing that's happening as well, which is that the signs alternates, 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 gonna have sort of combine some of the rules that I've seen before. So remember whenever you have alternating signs, it always means that there's gonna be a negative one to the nth power. And remember if the first term is uh is, is uh negative, that means that we're gonna have an N and not N plus one. So if the first term is negative, then it's just gonna be negative one to the end power. Now, obviously, this isn't just enough by itself because if I do negative one to the end power, that's just gonna alternate from negative one to positive one. So we're gonna have to figure out how to increase these numbers by two each time. So what do I do? Remember a lot of times you're gonna have to multiply constants or different things like that. So can I just multiply this whole sequence by two? Well, what happens here is if I just carry out this sequence and figure out the couple of numbers, I'm gonna have negative one, positive, one, negative one, positive, one times two. So in other words, it's just gonna be negative two, positive, two, negative two, positive two. So that's not enough either. Now, remember whenever you increase by the same number each time like plus one or plus two or plus three, that means that you're gonna have either N or two N or three N or so on. So in this case, what's gonna happen is we're gonna have to increase, you're gonna have to multiply this by two N. All right. So this two N is now gonna account for the fact that now for each term you're gonna increase by two 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. Uh So for a one, this is gonna be negative one to the one power times two times one. So in other words, this is just gonna be negative one times two, which is negative two. That's exactly what the first term is. Let's take a look at the second term, this is gonna be negative one to the second power times two times two. So in other words, this is just gonna be positive one times four, that's gonna give us positive four. That's exactly what the second term is. One more just for, you know, just to be safe, this is gonna be negative one to the third power times two times three. So in other words, this is gonna be negative one, right? Because it's to the third power, it's an odd power and this is gonna be times six. So this is gonna be negative six. And that's exactly what our third 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 to N and it'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 to N to the NTH power. It doesn't work like that. You can't like absorb something into that parentheses. All right. Um So you're just gonna have to leave it like this general formula. So now how would you figure out the 18th term? Very simple? We just plug in 18 in for the N or for the index in this formula? So the 18th term is just gonna be negative one, raised to the 18th power times two times 18. Now, if you plug this into your calculator, what you should see is that negative one to the 18th power is just one, right? Because it's an even power and then one times 36 this equals positive 36. So that's the general formula. And then for the 18th term we just have positive 36. All right. So thanks for watching. Hopefully, this made sense and I'll see you in the next one.

7

concept

Recursive Formulas

Video duration:

5m

Play a video:

Hey, everyone. So in previous videos, we saw how to use the general formula to figure out the terms of a sequence. So for example, if I had something like a sub N equals two N, then the whole idea was I can grab these indexes 12345, I plugged them into this equation here to get my outputs. What we saw is that the first five terms of the sequence were 2468 10. But I'm gonna show you this video is that sometimes you may be asked to use or write a different kind of formula called a recursive formula. I'm gonna 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 end 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 gonna break down the difference. I will 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 sub N equals two N, 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 sub N which is the next term in the sequence is equal to a sub N minus one plus two. With this really, with this A sub N minus one is, is, it's basically just the previous term in the sequence. It's the index N subtracted by one. So it's the previous term. So really, so let, let's take a look here, let's say you have this formula A sub N minus one plus two and you have the first term in the sequence which is a one, let's calculate the next few terms of the sequence using this formula. All right. So if I wanted to calculate a two, what this formula tells me is I'm gonna have to use the previous term in the sequence. A one. And I'm just gonna have to add two to it. All right. And so this a one is, it's just two. So two plus two equals four. Now let's calculate the next term. This A three, what this formula says is that to calculate a three, I need a two and I have to add two to it. We actually just calculated what two is, that's just the four that we just calculated. So I'm gonna have to take four, add two to it and now I get six. So now, I have four and six. If I wanted to calculate a four, you'll see the pattern here. I have to know what a three is and so on and so forth. So you're just gonna get six plus two and you're just gonna get eight. So if you continue on this pattern, which you're going to see is that we get 42468, 10 and notice how we have actually ended up with the same exact numbers that we did uh 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 gonna need N to plug in to 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 minus one to get what the next term is. That's the main difference 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 two So I'm just gonna continue on that pattern. So it's not that one is always better than the other. Just depends on what you're given it and ask, let's go ahead and take a look at some examples here and work this one out together. All right. So given this recursive formula and the first term of each sequence, I wanna find the next three terms in the sequence. All right. So I'm told here that this, in example, a ace of N equals two times the ace of N minus one. That's the previous term. And I'm gonna have to add three to it. And I'm already told what a one is, that's just equal to one. Let's use this formula to now find the second term, second term says uh or this formula says in order to find a two, I'm gonna have to take two multiply it by the previous number in the sequence, which is a one and I'm gonna have to add three, but notice that we've already calculated that it's just 20 sorry. A one is one is gonna be two times one plus three. If you work this out, what you're gonna get is five. So the second term of the sequence is five. Now let's take a look at the third one. This A three says in order to calculate the next term I'm gonna have to take two and multiply it by the previous term A two and add three. We've already figured out what A two is. It's just five over here. So this is just gonna be two times five plus three and that gives me 13. Now, for the fourth one, this is gonna be two times the previous number a three plus three, you've already figured that out. That's just 13. So this is gonna be two times 13 plus three and this is gonna equal 29. All right. 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 gonna be really, really good for this. All right. 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 seven N equals N times the previous term in the sequence. And I've got a one which equals one. What's the second term? Well, a two just tells me that it's just gonna be the index itself. In which case I'm using N equals two. So this is gonna be two times a one. So in other words, this is just gonna be two times the previous number, which is one and that gives me two. So that is a two. Um So now let's calculate a three. This is where N equals three, right? So what I'm gonna do is this formula says take the index which is three in this case and now multiply it by the previous number which is a two. So what is that? So this is gonna be three times two and that gives me six, That's the third term in the sequence. So now for a four, hopefully you see the pattern. Now this is where N equals four, you're gonna grab the index which is four multiplied by the previous term, which is a three. This is gonna be four times six and this is gonna be 24. All right, that's gonna be 24 over here. All right. And that's how you use these, uh, these recursive formulas to figure out the next few terms of a sequence. Hopefully, that makes sense. Thanks for watching and let's get some practice.

8

Problem

Problem

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