**Calculus 2 - Sequences & Series: What's a Sequence? โ€” Transcript & Summary | SozAI** Source: https://sozai.app/transcript/calculus-2-sequences-series-sequence/ Introduction to sequences, recurrence relations, and explicit formulas in Calculus 2 with examples and graphical interpretation. ## Key Takeaways - Sequences are ordered lists of numbers with a defined starting point and indexing. - Recurrence relations define each term based on previous terms, showing dependency. - Explicit formulas provide a direct way to calculate any term based on its position n. - Sequences can be graphically represented as discrete points, not continuous curves. - Understanding both recurrence and explicit forms is essential for analyzing sequences. ## What the video covers - Defines a sequence as an ordered list of numbers with a starting term and often a relationship between terms. - Introduces notation for sequences such as A1, A2, A3 or starting at A0, and the concept of indexing with n. - Explains recurrence relations where the next term depends on the previous term, e.g., An+1 = An + 3. - Describes explicit formulas where terms are expressed directly as a function of n, e.g., An = 1/(2^n). - Shows how some sequences can have both recurrence relations and explicit formulas. - Provides examples calculating terms using both recurrence relations and explicit formulas. - Demonstrates graphing sequences as discrete points rather than continuous functions. - Works through examples of identifying patterns, writing recurrence relations, and explicit formulas. - Highlights the importance of the index n and how sequences can be infinite. - Summarizes the difference between recurrence relations and explicit formulas and their applications. Answers ## Questions about this video What is a sequence in mathematics? A sequence is an ordered list of numbers, often infinite, where each number is called a term and is indexed by a counting number n. What is the difference between a recurrence relation and an explicit formula? A recurrence relation defines each term based on one or more previous terms, while an explicit formula expresses the term directly as a function of its position n. Can a sequence have both a recurrence relation and an explicit formula? Yes, some sequences can be described both by a recurrence relation and an explicit formula, providing two ways to generate the terms. ## Full Transcript โ€” Download SRT & Markdown 00:00 Speaker A Hello everyone, welcome back to Math with Allison. Today, we're starting a new series, so we're going to be doing sequences and series. So let's go and dive into it. Today, we're going over what is a sequence. So we're just going to be very general today, but we're also going to be talking about a recurrence relation and explicit formula. So stay tuned. So let's go to think about sequences here. Just think of a list. A sequence is literally just a list of numbers, but there are some things about sequences that we can talk about. First is that we always have a starting number. So notice here that we start with the value of one, right? We also can find a relationship very often between the numbers. So with this sequence, notice that what we're doing is we're adding three every time we increase. So notice we start at one and then we increase three, right? And then from four to seven, we add three. From seven to ten, we add three, and that relationship is going to keep going on, right? Another thing that we can talk about is that we can name the numbers. So very often what we do is we write this as a sub one. It is the first number of the sequence. The second number is a sub two, the third number is a sub three, so on and so forth. So a sequence looks something like this: A1, A2, A3, and on and on. Some people like to start at zero though. They say A0, A1, A2, A3, and so on. Another way that you can write this is A sub n, starting at n equals 1 going all the way to infinity because notice these sequences don't have an end. This dot dot dot means it's going to go on forever. And finally, we can write some kind of formula for the numbers. So notice here that I have that A sub 1 is equal to 1, A sub 2 is equal to 1 plus 3, which is equal to 4, A sub 3 is equal to 4 plus 3, which is equal to seven. But what if I went ahead and rewrote this as A sub 2 is equal to A sub 1, which is the one plus 3? Then I can also rewrite this one as A sub 3 is equal to A sub 2, which was equal to 4 plus 3. So notice what kind of relationship we have going on here. We have A n plus 1, which is the bigger one, is equal to A n plus 3. A n is a previous term, right? The reason it's n plus one is because notice here we have A3 compared to A2. Here we have A2 compared to A1. So this A n is always going to be one less. And so here we have our formula, and this is actually called a recurrence relationship. We'll go more into that, and we say for n is equal to 1, 2, 3, all the way down to infinity, right? And so this is the kind of relationship that we have. So that's a little nice introduction, but let's go through the definition. We have a sequence. The squiggle lines A sub n is an ordered list of numbers. Notice here that it's ordered, and it's of the form A sub 1, A sub 2, going all the way down and forever, right? We have the sequence may be generated by a recurrence relation of the form A sub n plus 1 is equal to F of A sub n. So in our case, we were A sub n plus 3. That's a function of A sub n. Think of the A sub n as an x, right? It's just a function of x. And this is for n is equal to 1, 2, 3, where the first term A sub 1 is given. So in our example, right, A sub n plus 1 is equal to that whole thing, and we also had that our first term was equal to one. Another way that this may be defined is with an explicit formula. So instead of being a function of A sub n, it's now just a function of n. And normally A sub n is equal to it, and then we have n is equal to 1, 2, 3. We're going to go ahead and see an example of this, and we've already talked about the notation, so I'll just leave that there for you guys to look at. So here we want to talk about an explicit formula. We have that A sub n is equal to 1 over 2 to the n, which is why it's a function of n. So here we want to go ahead and write out the first five terms of the sequence. So let's go ahead and do A sub 1 here. We're going to plug in n is equal to one, so we get 1/2. A sub 2, n is equal to 2, so we get 1/4. A sub 3, n is equal to 3, so 1 over 2 cubed, which is equal to 1 over 8. I'm going to move over here. We have A sub 4 is going to be equal to 1 over 2 to the 4th, which is 1/16, and then A sub 5, I'm just replacing n each time with the same index, right? And this is equal to 1 over 32. So if I were to write this out as a sequence, right, we have our first term. That was a terrible squiggle. That's okay. 1/2, 1/4, 1/8, 1/16, 1 over 32, so on and so forth. The little vocabulary for us, this right here, the little numbers, is called the index. And so the index is n. It's going to be counting numbers one, two, three, four. Sometimes we start at zero. It's going to be that sort of thing. So in summary, a recurrence relationship depends on the previous term. So in our previous example, A n plus one, the next term depends on A sub n, and we add three to it. Explicit formulas depend directly on n. So in our example, A sub n was equal to 1 over 2 to the n, and both of these have for n is equal to 1, 2, 3, so on and so forth, right? So notice in that explicit formula, there's actually a way we can write this in terms of a recurrence relation. So notice here that we're starting off A sub 1 is equal to 1/2, and each time what we're doing is we're multiplying by 1/2, right? Multiply by 1/2, multiply by 1/2, so on and so forth. So we would say A sub n plus one, the next term, is equal to 1/2 times A sub n. So it doesn't always have to be one or the other. It can be both. This has both an explicit formula and a recurrence relation. So let's try an example here. We have A sub n is equal to n times n over n squared plus 1. We're going to write the first four terms, and we're going to sketch a graph of it. Yes, we can graph sequences. And this right here is going to be an explicit formula. So let's go ahead and do this thing. A1, when we plug in one, is going to be equal to 1 times 1 over 1 squared plus 1. That's going to be equal to 1/2. So let's go ahead and graph that. We're going to go down here, and we're going to fill in our point. A1, it is just a point. So now let's do A2. That's going to be 2 times 2 over 2 squared plus 1. N is equal to two, right? So we get 4 over 5. This is going to be 4/5. So I'm going to go over to my two, right, because this represents the index n, and then our y-axis is actually the A sub n, right? So here for A2, we're going to go up to 4/5, which is probably going to be like right there. I'm just guesstimating. A3, we're going to go ahead and plug in n is equal to 3, which we can see on our little graph here. We get 3 times 3 over 9 plus 1. That's going to be 9 over 10. It's going to be a little bit above a quarter, and so we get A3. Let's go and try to graph out A4. So A4 is going to be 4 times 4 over 16 plus 1. That's going to be 16 over 17, and I'm just going to guesstimate again that A4 is going to be something like this. So notice this is not like a solid function. It's not going to do something like this. It's only filled in at these specific indices, right? And so it's just going to be a graph of points, and that's what a sequence looks like graphically. Here we have that given A sub n is equal to 2, 5, 12, 19. Let's go ahead and just do the first one. We're going to write out the next two sequences of the terms. So let's go ahead and try to find a pattern. So notice each time, right here I'm adding seven, right here I'm adding seven, right here I'm adding seven. So what I'm going to do in order to get my next term is I'm going to add seven. So we get 26. 26 plus 7 is 33, and those are our next two terms. For the second bullet here, we're going to find a recurrence relation that generates a sequence. So of course, for recurrence relation, we always need the first value, and that's going to be negative two. So here we have our next term is going to depend on adding seven to the previous term, right? And so that's a recurrence relation. And then here we're going to write out our explicit formula. So I'm going to rewrite our sequence. What I'm going to do is try to find a pattern. So here, and I'm always going to start with the first term. From the first to the second, I add seven. From the first to the third, I add 14. From the first to the fourth, I'm going to add 21. And then finally, from the first, what is that, to the fifth, I'll just stop with that one. I'm adding 28. So let's go and... 00:11 Speaker A to be very general today but we're also going to be talking about a recurrent relation and explicit formula so stay tuned so let's go to think about sequences here just think of a list a sequence is literally just a list of 00:22 Speaker A numbers but there are some things about sequences that we can talk about first is that we always have a starting number so notice here that we start with the value of one right we also can find a relationship very often between the 00:33 Speaker A numbers so with this sequence notice that what we're doing is we're adding three every time we increase so notice we start at one and then we increase three right and then from four to seven we add three from 7 00:48 Speaker A to 10 we add three and that relationship is going to keep going on right another thing that we can talk about is that we can name the numbers so very often what we do is we write this as a sub one it 01:00 Speaker A is the first number of the sequence the second number is a sub 2 the third number is a sub3 so on and so forth so a sequent looks something like this A1 A2 A3 and on and on some people like to 01:14 Speaker A start at zero though they say a zero A1 A2 A3 and so on another way that you can write this is a subn starting at Nal 1 going all the way to Infinity because notice these sequences don't have an end 01:28 Speaker A this dot dot dot means it's going to go on forever and finally we can write some kind of formula for the numbers so notice here that I have that a sub 1 is equal to 1 a sub 2 is equal to 1 + 3 01:41 Speaker A which is equal to 4 a sub 3 is equal to 4 + 3 which is equal to seven but what if I went ahead and rewrote this as a sub 2 is equal to a sub 1 which is the 01:53 Speaker A one + 3 then I can also rewrite this one as a sub3 is equal to a sub 2 which was equal to 4 + 3 so notice what kind of relationship we have going on here we have a n + 1 which is the bigger one is 02:09 Speaker A equal to a n + 3 a n is a previous term right the reason it's n plus one is because notice here we have A3 compared to A2 here we have A2 compared to A1 so this a n is always going to be one less 02:23 Speaker A and so here we have our formula and this is actually called a recurrence relationship we'll go more into that and we say for n is equal to 1 2 3 all the way down to Infinity right and so this 02:35 Speaker A is the kind of relationship that we have so that's a little nice introduction but let's go through the definition we have a sequence the squiggle lines a subn is an ordered list of numbers notice here that it's ordered and it's of the form a 02:48 Speaker A sub one a sub 2 going all the way down and forever right we have the sequence may be generated by a recurrence relation of the form a subn + 1 is equal to F of a subn so in our case we were a 03:00 Speaker A subn plus 3 that's a function of a subn think of the a subn as an X right it's just a function of X and this is for n is equal to 1 2 3 where the first term a 03:11 Speaker A sub one is given so in our example right A subn plus 1 is equal to that whole thing and we also had that our first term was equal to one another way that this may be defined is with an explicit 03:22 Speaker A formula so instead of being a function of a subn it's now just a function of N and normal a subn is equal to it and then we have n is equal 1 2 3 we're going to go ahead and see an example of 03:32 Speaker A this and we've already talked about the notation so I'll just leave that there for you guys to look at so here we want to talk about an explicit formula we have that a subn is equal to 1 over 2 to 03:42 Speaker A the N which is why it's a function of n so here we want to go ahead and write out the first five terms of the sequence so let's go ahead and do a sub one here we're going to plug in N is equal to one 03:53 Speaker A so we get 1 12 a sub 2 N is equal to 2 so we get 1/4 a sub3 n is is equal to 3 so 1 over 2 cubed which is equal to 1 over 8 I'm going to move over here we 04:05 Speaker A have a sub4 is going to be equal to 1 over 2 to the 4th which is 116th and then a sub5 I'm just replacing n each time with the same indic right and this is equal to 1 over 32 so if I were to 04:19 Speaker A write this out as a sequence right we have our first term that was a terrible squiggle that's okay2 1/4 1/8 116th 1 over 32 so on and so forth the little vocabulary for us this right here the little numbers is 04:33 Speaker A called the index and so the index is n it's going to be counting numbers one two 3 four sometimes we start at zero it's going to be that sort of thing so in summary a recurrence relationship depends on the previous term so in our 04:46 Speaker A previous example a n+ one the next term depends on a subn and we add three to it explicit formulas depend directly on N so in our example a subn was equal to 1 over 2 to the n and both of these have 4 05:00 Speaker A n is equal to 1 2 3 so on and so forth right so notice in That explicit formula there's actually a way we can write this in terms of a recurrence relation so notice here that we're starting off a 05:12 Speaker A sub 1 is equal to 12 and each time what we're doing is we're multiplying by 1/2 right multiply by 1/2 multiply by 1/2 so on and so forth so we would say a subn plus one the next term is equal to 12 \* 05:27 Speaker A a subn so it doesn't always have to be one or the other it can be both this has both an explicit formula and a recurrence relation so let's try an example here we have a sub nal1 n \* n / 05:40 Speaker A n^2 + 1 we're going to write the first four terms and we're going to sketch a graph of it yes we can graph sequences and this right here is going to be an explicit formula so let's go ahead and 05:49 Speaker A do this thing A1 when we plug in one is going to be equal to1 to the^ 1 \* 1 / 1^ 2 + 1 that's going to be equal to1 12 so let's go ahead and graph that we're 06:02 Speaker A going to go down here and we're going to fill in our Point A1 it is just a point so now let's do A2 that's going to be 1 s which is positive 1 \* 2 N is equal to 06:12 Speaker A two right so we get 4 + 1 this is going to be 2 this so I'm going to go over to my two right because this represents the index n and then our y AIS is actually the a subn right so here for a two we're 06:27 Speaker A going to go up to two fths which is probably going to be like right there I'm just guesstimating A3 we're going to go ahead and plug in N is equal to 3 which we can see on our little graph here we get -1 \* 06:38 Speaker A 3 / 9 + 1 that's going to be -310 it's going to be a little bit above a quarter and so we get A3 let's go and try to graph out A4 so A4 is going to be 1 \* 4 / 16 + 1 that's going to be 4 over 06:55 Speaker A 17 and I'm just going to guesstimate again that A4 is going to be something like this so notice this is not like a solid function it's not going to do something like this it's only filled in at these specific indices right and so 07:08 Speaker A it's just going to be a graph of points and that's what a sequence looks like graphically here we have that given a subn is equal to2 5 12 19 I let's go ahead and just do the first one we're 07:19 Speaker A going to write out the next two sequences of the terms so let's go ahead and try to find a pattern so notice each time right here I'm adding seven right here I'm adding seven right here I'm adding seven so what I'm going to do in 07:33 Speaker A order to get my next term is I'm going to add seven so we get 26 26 + 7 is 33 and those are our next two terms for the second bullet here we're going to find a recurrence relation that generates a 07:46 Speaker A sequence so of course for recurrence relation we always need the first value and that's going to be -2 so here we have our next term is going to depend on adding seven to the previous term right and so that's a recurrence relation and 07:59 Speaker A then here we're going to write out our explicit formula so I'm going to rewrite our sequence what I'm going to do is try to find a pattern so here and I'm always going to start with the first term from 08:10 Speaker A the first to the second I add seven from the first to the third I add 14 from the first to the fourth I'm going to add 21 and then finally from the first what is that to the fifth I'll just stop with 08:22 Speaker A that one I'm adding 28 so let's go and try to rewrite these this is going to be the from the first to the second we're going to take -2 and we're going to add 7 \* 1 from the first term to the third 08:36 Speaker A we're going to add 7 \* 2 right we're adding 14 from the first to the fourth we're adding 21 which is equal to 7 \* 3 and then finally we're taking -2 and we're adding 28 which is 7 \* 4 so our 08:50 Speaker A explicit formula is going to look something like this a subn is equal to -2 + 7 \* n for n is equal to and actually what we're going to be starting at is zero in order to get the first 09:01 Speaker A term right because we have to get the -2 if we start at 1 we're going to get -2 + 7 which is equal to 5 which is the second term so that's why sometimes we start at zero which is totally okay to 09:10 Speaker A do that and then we're going to go to one two three so on and so forth okay finally we have another problem we're going to be doing the same thing but we have a new sequence so here we have 3 6 09:20 Speaker A 12 so on and so forth so for our first thing we're going to go ahead and find the next two terms of the sequence so I'm going to rewrite it let's go ahead and find the pattern so for the first to the second I'm 09:31 Speaker A multiplying by two from 6 to 12 we're multiplying by two 12 to 24 we're multiplying by two we're beginning to see the pattern here so let's multiply this by two we get 96 multiplying that by two we get um 09:43 Speaker A 192 and those would be the next few terms of our sequence so now we're going to go ahead and write out a recurrence relation that generates a sequence so here we have the the first term is going to be equal to three and now we want to 09:57 Speaker A figure out a way to find the next term depending depending on the previous term and that's going to be taking the previous term and multiplying it by two right or you can rewrite that as 2 a either of those are going to be totally 10:09 Speaker A okay finally we want to write an explicit formula so let's go ahead and rewrite the sequence so remember just like last time with adding I want to find a relationship between the first term and all of the following this one's not 10:23 Speaker A going to be adding though because we now we have a sequence that's being multiplied so now I want to think of it in terms of multiplication so in order to get from 3 to six we multiply by two right in order to get 10:34 Speaker A from 3 to 12 we need to multiply by four to go from 3 to 24 we need to multiply by 8 and then finally to go from 3 to 48 we're going to have to multiply by 16 right so let's go and try to rewrite 10:48 Speaker A this so in order to do six we multiply 3 \* 2 right in order to get to 12 we multiply 3 \* 4 but that can be Rewritten as 3 \* 2^ 2 for for 24 we took three and 11:01 Speaker A we multiplied by 8 but let's rewrite that as a power of two because we can see a pattern going on here that's going to be 2 cubed finally we took three and we multiplied it by 16 and again I'm 11:11 Speaker A going to rewrite that as 3 \* 2 4th so notice here what's changing the thing that's changing is going to be what our n is equal to everything else that's the same so three three three three that's all the same we have times 2 that's all 11:24 Speaker A going to be in our normal formula so here we have that a subn is equal to 3 \* 2 to the n and this one has to start at zero as well right because we have to get the first term if we plug in zero we 11:37 Speaker A get 3 \* 1 which is equal to three if we started at one we would be missing three but we need to have that as our starting term so that's all I have for us in this video today if you enjoyed it I have 11:46 Speaker A many more like it so make sure to check out my playlist our link down below otherwise please give this video a thumbs up and comment other problems or topics you'd like to see them thanks for [Music] watching [Music] Topics: Calculus 2 Sequences Series Recurrence relation Explicit formula Mathematics Math with Allison Sequence notation Graphing sequences Infinite sequences --- This is the markdown twin of https://sozai.app/transcript/calculus-2-sequences-series-sequence/ โ€” the same content, without the markup. 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