Hi miss hi am. How are you today? Excellent. So last time we were looking at series, so what I wanted to do today is go over a little bit more theory and see if we can put it all together in some questions. But before we dive in, is there anything that you have any questions on once youmold ded over? Any questions from your side? No. Okay. So the little bit of theory I want to go over first is something called sigma notation. Is that something you've heard of just to see where you're at? Yeah, okay. So so that I'm not lecturing you, can you tell me how far you've got with that? Is that something that you understand a bit about or you've just heard of it? I know some of it. I don't know of the equations, but I do know how how the how it works and what it expands to. So Yeah, it's really it's a notation, not a technique. It's just a new way of writing sums. It's especially useful when we can see what the pattern is. So for example, we've been going so far with dot dot dots, right? So something like. Something like that. But when we can see the pattern, we can write it in a sigma notation. So for example, can you see what the pattern is here in this sum I've written down? Three in minus two. No, quite. It's not arithmetic, actually. It's A I chose one that's a bit more immediate, okay? It's not arithmetical, geometric, actually. Can you spot the pattern? Ends with two squared minus one. Not quite. So if you think about the n numbers. So we've got one, two, three, four, and then I'll give you a clue. This is the tenth. Can you see what each of the top numbers correspond to? N, Yeah, so I'm adding up the squares so so that I'm not lecturing you. Things you already know. Just have a go. Could you know how to put this into sigma notation? Not a problem. If not, I just want to see where it went. Oh, yes, of course, always forget. Got that. There you go. Okay, almost. So I see that you are no hundred. So excellent. So I see that you understand that we put the kind of pattern in here and this end that you've done here, this is what we call a dummy variable because it doesn't mean anything in the abstract. It's just what we're plugging in each time. And our numbers here and here tell us which which numbers to plug in there. So one up to ten, excellent. So it seems like you've already got a pretty good understanding of that. Maybe just a few more to check that you're really happy what's going on there. What about I'm not going to do geometric and arithmetic for now. I just want to see that you could what about. And this time, let's say forever plus dot dot dot just keeps going. Could you put that in sigma notation? Yep, nice. Very nice. And let's just see if you can go the other way. What about. Do write that out in sort of dot dot dot notation or just write out the complete sum because it's quite short, isn't it? Very nice. Yeah so we're pluantime excellent. I think that's all to say about that. Maybe just to comment that it's really important we understand this dummy variable, as it's called, doesn't have a particular value. That's just what we're checking through each time we're going through all those values. But that's all I really wanted to Yeah say about that. So now that we've got a bit of the theory covered, I'd really like to make sure we've got a nice deep understanding by doing lots of practice, and I'll try and make these questions harder and harder. So keeps you interested. Let me just send you a worksheet. And of course, this is quite a formula intensive topic. And just to comment that I'm really not fast whether you've memorized them at this point, my dream scenario at this point is you go, Oh, and then there's a formula and it involves something like this, this and this. That's a really great place to be because it means you know what's in your toolkit. You know there's a formula involving these terms. If you haven't memorized them at this point, I don't mind. So I wonder if I can get it up in the yes, that's quite good. So looking at this sheet, let's start. Let's start with question two. Let's do two a. Could you give that a go for me? So two M5. That's correct. Very good. And yet we've written that out and segment nicely. So Oh yes, that's what they've asked thought in this topic. Nice. So let's then jump across two. Now question three again, let's just use three a. We don't need to get fiddly numbers involved. That doesn't help anyone. Three A, I can't see your browser though. It's blank. So I'm just going to look at my screen. Oh, okay. As long as you can see it one or another, I don't mind. But I don't know what the sum formula was not a problem that we haven't memorized this at this point. I think they even give it to you as a formula at a level, but it's good to have eventually internalized. I'll just here crucially, you're realizing that this is a moment when we then defer to the formula that we proved together, didn't we? And can you remember what an and d stand for here? I think n is the total number of terms, a is the first term and d is. The last term, not quite so. The only one though we need to remember d is our common difference. So the constant that follows. So yes, but thinking about it more kind of orgically. Maybe it's the difference between each pair of terms. We're going up by four each time here. Nice. Okay. So give it a go. The. I can scribe for you if it's too hard writing on the board. Oh, it's okay. Think about that 156. Did you think about that? Yeah nice. And then three nice. Just a comment. I mean, doing these sums is not the interesting way, but just a comment that if that there was a non calculator paper, we could make our life easier by taking a factor of two hours, couldn't we? That might be Yeah might be helpful, but that's not what I'm interested in. Good stuff, right? There's kind of more interesting one going. Skip ahead of it. Miss, what would the common difference be if it was a quadratic? Well, if by quadratic you mean a term that has a squared term in the nth term, is that what you mean? Yeah. Well then it wouldn't have a common difference, would it? Because the difference will not be the same between each two terms. Yeah. Yeah, that makes sense. Yeah. So it has another formula. Yeah. So there's a different formula completely because we don't have that common difference. Yeah. Okay, let's do something more interesting. What about. Look at question nine. I have done that. Question nine, let's have a go at that one. It says to write down two equations of. Yeah, we want some equations with a and d, and my policy still stands. If you can tell me that there's a so my policy is, if you can tell me there is a formula that tells me this, then I will give you the formula. That's my policy at this point. So is there a formula that, you know, exists that youlike me to quote for you? So I've given you one in the corner. I don't think that one's going to be very useful here. Do you remember vaguely what the other formula we encountered was? So that was the sum of the first n. So kind of simpler than that is just what's the nth term? We have a formula for the nth term, which is our starting number plus m minus one. I would say that's not even really one to memorize. That's quite intuitive, right? Because we say we have our starting number and then we have n minus one, lots of the differbecause. Each time we're going on, we're adding one of those differences. So that's nice. We don't even really need to memorize that. So there in your toolkit, they are really you're only two formally or arithmetic slide caveat, we have a second version of this one. You can also see a plus l, but with that caveat, they're really the only two formula you ever need for arithmetic. Okay. So with that in mind, have a go and really write some equations for a and d. Does that does that the equation work? That's correct. Let me just think, Yeah, that's correct. Probably not quite what they had in mind, but I'll accept it. The only thing is we also need to get a involved, don't we? So that is a correct equation. As long as you can now find me another equation that has an a, then I'm happy with that. So I need to give another equation. Well, it says we need equations relating a and d, and so far you've only written an equation with d in it. So I'll accept that as a correct equation, absolutely, but we need to get an a equation going as well. Okay, tell me about that change you've made. I don't know. Yeah, I don't think always assume I can tell when you're guessing that was a correct equation that you wrote down. Absolutely. And it's fine to do something different from what I reckon they meant when they set this question. The only thing is that you were yet to come up with a an a equation. So you absolutely could have stuck with that original one. We just need to find a way of relating a. So try and think really strategically here. Look through a little tobox of equations and see where a appears. What are some equations we could use that involve a, and how could we relate them to this scenario? I don't think I know any. Okay, so I've written down three equations in that little box in the Cora. So all you have to do is go through literally just three options and see whether they look plausible. So the first and the last, so the top line and the little last direcx I've written, they're both equations for the sum of the first n terms. Now does that seem relevant to this question? No, no, we're not really asked about that at all. Well, that leaves just one option. So if any of these equations are useful, it's going to be that middle one, right? So see, you can use that un equation because it has an a in there. So that's looking promising. See if you can apply that here. How could we use that middle equation, that un equation? Also it just wants me to substitute Yeah. Absolutely. Nice. 没。Very nice, and now I recthink you can have a good part b. Excellent. Very nice. So now that final part, have a go for the fortieth term facseries. And then imagine when you have a calculator that fine with that. Excellent. Very nice. Next, I would like you to have a go at question ten. Nice. So Yeah, that's what I was looking for. You've seen the you've taken away the message from question nine. Very good. And just tell me how you do part b and then we can move on something ghtly more interesting. I just use the formula on the top right corner. Yeah, spot on. Excellent. More interesting question, I think, is question twelve. Have a look at twelve. I mean. You need a refresher on what a natural number is. That's fine. Just integers, positive integers. Yeah Yeah. Sometimes zero included. Sometimes it isn't. That's kind of a matter of convention. Luckily it doesn't really matter here. So think of it as by natural it means normal numbers you come across in your life, something where you could have this amount of pairs or barrikets, you know first numbers that we needed to use as humans. Okay, thank. Excellent. And I can see that you've seen how gso, it's just an arithmetic series with Yeah but so they suggest that the starting term is zero event. No, that would work because it's we want nn plus one yet. So two works there, doesn't it? Oh, Yeah, Yeah. Yes, I would go with assumstart starting with one unless less told otherwise. Excellent. So nice. And part b, something a little interesting going on there. That Yeah you've spotted the little trip fair we need to subtract to excellent Yep. And then we'll plug them in. Yeah Yeah give that a go. Yeah you've spotted the interesting thing there. And then imagine we have our calculated to hand. We don't need to waste time with that. Excellent. Really well spotted. How we can do sums not starting at one by using subtraction. Very good. And Yeah whatever that comes out as excellent. Have a it now for me at 14b. So I like what the look of this. It looks like you're using that first equation. What do we need to work out before we can use that equation? What are the ingredients we need to put into the difference, not just the difference, right? I mean, some of these are office, but we we need n, we need a, we need d. Yep. So if we start between a, which Yewe can just plug in that, so that's all done excellent and it's and kind of easy, isn't it? And d, Yeah, we can just see by the fact we're minus into each time. Just a little point though, is d positive or negative here? Negative, Yeah, absolutely nice. Give that I go see what that spits out. That's looking good to me. That's not quite right. That should be right. If you plugged in. And you've definitely done it with the minus two. Yeah, that's not quite spitting out the right number. Let me just check that soon. Yeah just just check that's the right sum but it's not spat out the right number so just check you plugged that in right? I've plugged that sum into my calculator. It's giving the right answer. So just check your calcuated technique there. Okay. Nice. Well, that's seeming really confident. I think we can probably skip on to geometric series. Let me send you the worksheet for that. Yeah, I have. There you go. And against same policy, you can tell me that there is a formula that's kind of like this. I can put it in a little box at the top because I'm not interested in memorizing formula at this level, this point. I just want you to be aware of which formula we have is, I think, a sort of n plus something divided by n. So but what I mean by what formula we have is what's giving us. It's a formula for what? Yes, so a formula for some. And Yep, you're you're on the right tracks there. So for a geometric series A R to the n minus one or minus one, we don't need to do them all at the beginning. But just worth noting, there are two other important formula that we might need at some point. We can do that as we come to them. Let's start. With two b. So we can always test ourright, try plugging in a few values and see whether that's right. So your first step was really good. You worked out that minus four. Can you tell me what that minus four is? What's the significance of minus four? It has to be multiplied within. So to be precise, we multipeach term by minus four to get to the next one. Do you remember the special name we give that. Or the letter that we usually give that. We're not quite in geometric mode now, so we're not talking about the differences with little d. Remember, we're in geometric mode. So to be fair, we should have done this to start in this formula. In the corner we have ar and n. Can you tell me what all of those letters stand for? It is the start, the first term, and is a total number of terms. And R would be multiple. So almost can you tell me can you describe it? The amount multiplied, the term, one term to get to the next. Absolutely. We call this the common ratio, which means precisely that. How much do we times it by to go from ones leso? What you've done brilliantly there is you've worked out the common ratio minus four. The question is, how do we use that to get the nth term expression? So we've been using this U underscore n notation. How do we use the R to give us the un? The nth term, I'll give you a hint. It's not just multiply by n. And I'll give you another hint. There might be a formula you need. And if you can tell me vaguely what it is, then I'll give you the exact formula. That's my deal. So can you think of vaguely which formula would help you at this point? Not fast if you've exactly memorized it. Any ideas? So what I mean by that, that might not be totally clear. What I mean, this sn formula, I don't care if you've memorized it at this point, I don't care. What I would like at this point is you to be able to say we have a formula for the sum of the first n terms and it involves ar and n. That's a really great place to be at this point. Similarly, we have a formula for the nth term and it involves ar and n. That's where I want you to be right now. Eventually, we will memorize all of these, and I'll give it for you for now. Yeah and a bit like the nth term formula of four, it's pretty intuitive, right? Because we're starting with a and then times ing by R H times. So we're times ing at n minus one times. So pretty intuitive. We're definitely going to need that for this question. Have a go. Now that formula. Absolutely. It's easy to get caught out with not doing brackets. Really important. We have brackets there because that mine necessarily important. Excellent. Then all we have to do is fill in what that a is. Can you remember what a stands for? Excellent. Yeah, nicely done. Brilliant. Next up. Yeah, can you have it a three b? Yeah. Yevery good. So there's sum of the first twelve turns. And then we just plug that into our calculator. We don't necessarily need to do that now. Excellent for this sum. Does the sum to infinity exist? Little bonus question for you. Could I remove the browser for a second? Bear in mind, I'm just asking, does it exist? I haven't actually asked. What is it? I just want to try to maybe solve get it. Okay. Well, I'm interested in you being able to quickly tell me whether it exists or not, and then we'll think about what actually okay, well, if you knew that all along, I'd invite you to answer the question you're being asked. How did you know it exists? Because it's a fraction and so it should limit, it should be a finite value. Exactly. Can you tell me exactly can you make that more precise? What's our criteria? What tells us that it's going to exist? The fraction is a bit gonna go to some number infinitely large. Yeah but what you've said is it's a fraction. I'd like you to use precise language. What exactly is a fraction? N is the denominator of a fraction. I mean, not n is a part of the denominator of a fraction. Not quite. So the criteria we have is about R is the R value that tells us ours R is smaller than the absolute value of R is less than one. There we go. That's really important. That's our criteria. We can just look at it and instantly know, does it exist? Cool. But absolutely, my follow up question is going to be and what is that value? Miss, I'm just curious, what is the limit of n as it tends like any numbers, the power pin that would be infinity if the number is bigger than one. So you're two here. If that number anything to the power of n? If this thing here is less than if this thing in the box has a magnitude less than one, then it's going to be zero. And if it has a magnitude greater than one, then the limit doesn't exist. That is to say, it goes off to infinity. Can we just say that it's n. Say that what is n. Equthen, no, because things go off to infinity in different ways. And would just think about the graph, right? Do you know how we would draw the graph of. You know what that graph would look like. Yeah and that's very different from the graph of y equals x, isn't it? So they're both going off to infinity, but one's going off to infinity way quicker than the other. So they both don't have a limit, but that doesn't mean they're equal. And that's important because if we had something like if we just think, Oh, they're basically the same, then how on earth would we make sense of this opnote n on the bottom? You see what I mean? If we think, well, they're both about the same, then wehave something over the same and that give us one. But that's not correct. We have to remember that this thing on top is growing way quicker than the thing on the bottom. So what do you reckon? What do you reckon that limit would be? So not quite. You've done you've confused two things there. So the exponent laws, this is right. But we can't divide. We don't just subtract off whatever's on the bottom. Yeah, I think you'll miss quoting that. Yeah but we don't need so the point here is that the tops going to infinity, the bottom's going to infinity. If the tops go into infinity more quickly than the bottom, then it's overall going to keep getting bigger and bigger. If the bottom is going to infinity quicker than the top, it's going to get smaller and smaller. Yeah. So that's where it's really important to know that two to the n is growing much bigger than n. So this thing, actually the limit doesn't exist, goes to infinity. And then if I flipped it, can you tell me then. Yeah, exactly. So just because two things are going off to infinity, it's really important that we know how are they going off to infinity because we might want to combine them in interesting ways. Nice. That will make sense. Any other general limits questions? For that one, can we just if it limits to zero, can we just do its derivative? So one and n two n minus one, are you thinking lopatel? Yeah, Yeah because Yeah so because it limits the zero, we can't do that. So lopatals rule Yeah works when they're both going off to infinity. So that does work. However, you haven't differentiated the bottom correctly there. So Yeah, try not to get ahead of yourself. You're right or patalrule would work there. But much simpler just to know that two to the n is growing much faster than n. As a side point, that's not the correct derivative of the bottom. What you're thinking of, if we have a variable in the bottom and we differentiate that with respect to x, then yes, we bring down the power one. But if we have just some number, not quite if we have a number at the bottom. That's the same way of writing. Yeah, that's writing it another way. Yeah. So you're still that that's making a whole extra mistake. You're combining a lot of a level maps here to whithrobecause. I don't want to just say it's something you'll learn later. To differentiate this, the whole thing, we would have to use the product rule because we've got two products times by each other. The first issue was if we're just differentiating the top and the bottom, then you'll right the top differentiates to one, but the bottom doesn't differentiate to this. That's not right. The reason for that is the rule that you're quoting is we move the power to the front to the minus one. That's how we differentiate x to the n. What we're doing here is two to the x. Do you see how the variable is in the exponent, not the base? Yeah. So that's not quite right. This is actually quite a complicated thing to differentiate, which we can go over if youlike, but I feel like we're involving a lot of different techniques at this point. Okay. I realithat was covering a lot of ground. Let's, unless you have any other questions, go back to this initial question of what's the sum to infinity of this sum that I've now totally forgotten what it is this one. Yes, just finish that off. You've told me the sums invenexists. Can you tell me what it is? So the interesting part is, when you've gone from here to here, can you explain how you've gone from there to there? So this simplifies to zero Yeah, that's the main point by doing it from scratch. But can you tell me what the formula for the sum to infinity is where it exists? So what do you mean? So you've done it from scratch, which is absolutely fine, but can you tell me what the general formula is for the sum to infinity? And you don't need to remember it because you've basically just reproved it. And the sequence here, Yeah, but it simplifies down. Yes, but there's a closed form formula where we can just plug in arr, can you remember? You've kind of done it implicitly. You did it from scratch, which is fine. And that shows me that you could work out the formula because you thought about our sum to infinity formula, and then you sent n off to infinity. You did it with a particular value of a and R, but you should be able to do it generally as well. What's our general formula for any anr value using exactly the same technique you did? So basically, can you evaluate this limit just generally you did it specific case, but can we do it for general Anand R now? This is all assuming that R has modulus less than one. So do exactly what you just did, but now we're assuming okay, I was assuming that R can be any value. Well, it will only exist if R has modulus less than one year. Yeah. So aogara. Almost. So I like what done on top because you've realized this whole thing g's going to zero. Why have you got rid of the R, the minus one on the bottom? Yeah, Yeah kind Yeah. I was thinking about the specific case, which I don't know why I thought about that. Nice, so absolutely fine to do from scratch. And certainly we don't need to like memorize this because we can once we know how to do it, we can quickly see how it follows. But we do have this general formula, a over one minus R. Nice, let's just do. Oh, let's get p to the final question. Let's try and stretch you 20b well, 20 but with a view to doing 20b. This is question 20 we're looking at, which you might not quite find. Of the series. So okay. Yeah, so. The very final question. Yeah Yeah. So it's the rotation is this k depends what you're trying to denote. What are you trying to write down? Because the series k plus ten k and k minus six. So Yeah, they represent the number of term no, no. So so this is some case is some number and this forms a geometric series. So k has some money that we're going to find out. Curwith your notation s without a subscript usually denotes the sum if you're trying to denote the term you sorry, we tend to use you Yeah. Yeah, exactly. So be careful with that last thing you've written down. What the ten zero minus six, what does that mean exactly? I'm not sure how to get it. I was just trying to. Subject chat key from each term. Yes. So beware of guesswork. We do best in maths when we each step is a little step that we know is right and we build up from those little steps to a big result. But always best to do little things that you know work. So I like what you were starting to do youwritten. Now that's really great. What will make more useful to you is if you tell yourself what that actually is, what does that represent? Answer, Yeah, once we've done that, that suggests some things we could do because you know what the second and the third term are. You've kind of you've used this to put it in here, but we can now use these two. We're just about out of time. Feel free to try and finish this off for homework. I'll take a pick. C, so I remember where we've got to but have a play around with that. Hopefully thatit uses just the poor knowledge we've discussed, but it will hopefully stretch you to apply in an interesting way next time. I wanna finish off that question, but then I don't have any particular preference what topic we look at next. Are there any topics that you're raring to get into? I don't know. Limits, maybe introduction into calculus. Okay, Yeah, let's do calculus. Actually, that's a good idea. Yeah, cool. So next time we'll finish off this question and we'll see how you got on with it and then we'll start calculus. Great stuff. Excellent. Really great stuff today. Bye.