 Any questions you want me to go over now? It's your chance to ask. 3b, find the number of terms in each sequence. Okay, so here's a nice twist. They want us to find n. So I'm going to use this, Megan. Where it looks like the nth term is negative 1. It looks like a is 512. It looks like r is negative a half. Is that right? To the n minus 1. Where is the n sitting? An exponent. Oh, I'm not going to take the log of both sides yet. We said you always wanted to get the exponent by itself first. Remember that half-life question? We did that all the time too. In fact, I think in half-life we also call this a0. Anyways, almost the same idea. I'm going to solve this equation. Negative 1 over 512 equals negative 1 half to the n minus 1. Did you get an error when you tried to solve this? Okay, we may run into a negative issue. Because you can't just log a negative. Yeah. Now, we can say this. If this thing works out to an odd number, like if n is 10, n minus 9, if n is 12, n minus 9, 12 minus 11, you'll have a negative divided by a negative, which is a what? Positive. We can probably ignore the negatives. You tried taking the log of both sides and your calculator when you went log of negative 1 over 512 gave you an error. Yes, I just noticed that now and I went, oh, by the way, if this was a multiple-choice question, would you solve it this way or would you just try plugging in all four answers into there? We'll make sure we're clever because it is often a multiple-choice question. So what I'm going to do is I'm going to say this is a negative divided by a negative as long as this is odd. And you know what? It has to be because otherwise I can't get a negative answer. I'm going to assume that the negatives will cancel. If I do that, can you take it from there? Good. So I'm going to solve 1 over 512 equals 1 half to the m minus 1. Take the log of both sides, divide by log of a half, plus 1 and is 2, 4, 8, 16, 32, 64, 512. I think n is going to be 10 because I want a 9 up there if I've done my counting exponents right. It's a two-times table. That one I got memorized. Is that okay? Good for bringing that up. I've forgotten that obscure one. I suspect they deliberately won't throw one like that on the provincial. I doubt I did on my test. I don't think I did. That's a curve ball within a curve ball. I don't like those. Any others? So let me just say I like number eight, for example. I like number nine, this whole algebraic one, where they give you three terms with x's. And you've got to then figure out what r is by cross-multiplying and letting them equal to each other. And it's a quadratic. I get two answers. My little heart goes pitter-patter. And people get number 11. First of all, I'm going to give you a hint. It is geometric. The real question is what's r? How can I always find r? Yeah. Sorry? The answer D? Really? Mr. Dewick, is it? Let's see. Any term divided by the one in front of it. So log of x squared divided by log x. Move that to the front. And you get the logs cancel. Yay! And you get two. Oh, you know what? It is not a geometric one. I stand corrected. You're right, Megan. Because in the first term, r is two. And the second term, r is three. They do love to throw a logarithmic r in there, though. Trust me. Well, then let's move on. Turn to lesson two. Kim, all we're going to ask ourselves today is the following. What if instead of listing each term separated by commas, what if instead of commas, I want to put plus signs and add them up along the way? Is there a shortcut? And the answer is, yeah. Absolutely. So a series. If you take a sequence separated by commas, an amrit, instead of separating them by commas, you say add them up along the way. Okay? That's a series. A sequence separated by commas. A series, add them up. We're not going to ask you to remember which one is which, but we will use that terminology. So you need to recognize it when you see it. And there is a formula. And there is a lovely, lovely, lovely proof. But in the interest of time, I'm going to cut corners and I'm going to say, here is the formula. I'm going to do a box around this and I'm going to explain what everything means. First of all, Adam, is this on the formula sheet? Okay. Here's what it means. The letter s stands for the sum of. So s6 would mean the sum of the first six terms. What would s8 mean? The sum of the first eight terms. What would f infinity mean? We're actually going to look at that. We're going to say, sometimes you can add an infinite number of terms. Next lesson. Jan, you answered this already. What does a mean do you think? Yeah, still is. And what's r? What's r? I may call it the common ratio, because that's the fancy-schmancy term. And I want you to get used to seeing it, because they'll always write it that way. Yeah, the ratio. And, oh, what's n? The number of terms. So in other words, even though this formula looks ugly and Nicole, you'll have to memorize it. We've seen everything but the s. Sorry, I had a sneeze waiting there. There is a second version of this formula. But, Tally, I use this formula here. Oh, by the way, why does it say r can't be one? If r was one, what would you have in the denominator? Zero. But also, if you were multiplying by one, would the numbers change at all? Or would it just be the same number over and over? Yeah, it's technically not a geometric sequence. It's boring. Forget it. There's a second version of this, Jan. I'll use this first one if I know these, or I know three of these, and I need to find the fourth one. Sometimes, instead of knowing n, instead, I'll know the last term. That's why there's a little letter l there. Now, I think on your actual formula sheet, they do a cursive like a loopy l instead of a straight vertical weird one like this. But this is also on your formula sheet, and it says, oh, by the way, if you don't know how many terms there are, you can still add them up if you know the last term. It's the first term minus r times the last term divided by one minus r. You know what? It's much easier just to start using this, but you absolutely want to get your calculators out because nobody tries to do these in their head. Not even me. Example one. It says, determine the sum of the first seven terms in that sequence. Do I know the last term, or do I know how many terms? I know how many terms. I'm using the first one. Let's do our little physics defect approach. We're going to list some data here. So Nick, what's that more specific in this? I don't know. You have it in front of you. Trust on example one. Thank you. I know you're zoning out. Bring it your back. Okay. Flipping pages. Uh-huh. Good recovery. Almost smooth. Hey, let's keep going. Nick, that was so much fun. What's R? Can you tell me? Times by negative two, right? What's N? How many terms are we talking about? A bit trickier, Nick, because they wrote it instead of writing the number. But how many terms are we talking about? Seven. Are you really going to Vitaly for advice there? Because if you are, I mean, that's a pretty dry well to be dipping a bucket of water into. I'm just going to say that, okay? And even the water that does come out of it. Oh, yeah, be a little. What do they want us to find then? They want us to determine the what? You know what? They want us to find S7. And even though the formula is right at the top of the page, I think it's worth writing it out at least once so you can see how it all fits together. S7 equals A bracket one minus R to the M all over one minus R. The sum of the first seven terms without having to bother adding them up by hand is going to be equal to A, three bracket one minus negative two in brackets to the M, seven close bracket all divided by one minus negative two. I strongly encourage all of you to try this on your calculator right now because I found from prior experience, Evan, the mistakes are not math. They're calculator based. How would I type this in? I think I'd put brackets around the entire top. So I'm going to go like this. Bracket negative three bracket one minus bracket negative two close off the negative two to the power of seven close off this bracket close off the top divided by how many terms do I have in the denominator? Now I could go one minus minus two which is three but just in case it wasn't one that I could do in my head I would go bracket one minus negative two or if I was doing the same question over and over I knew they just wanted me to change one number at a time then you type everything and go second function enter and just change whatever they want you to change. What do you get? I think the answer is negative 129. Not a bad shortcut. Okay. Don't clear your calculator. You probably already did. Ah. Example two. Which formula am I going to use for example two? Did they tell me in example two how many terms there are? No. Did they tell me the last term? I'm going to use this one. S equals and this one I don't have memorized. There it is. A minus R and I always write L for last term as a cursive L instead of a printed L because what number does a printed L look like? A one and I don't want to think it's a one accidentally all divided by one minus R. Let's see Amy if we can go straight to plugging in numbers without making a list. Amy what's A according to this question here? Yep. Minus. What's R? You're right. I think say it louder. How can you always find it? Take any term and do what? Divided by one in front of it. I hope you went that divided by that. Not that divided by that. You get the same answer but I'll pick the easier ones. Right? So negative three. What's L stands for? L stands for last term. Which is negative 8748. All divided by one minus negative three. And again I'll write this. Amy what is one minus negative three in the bottom really? What's a minus minus the same as? Yeah okay so if you want to what could you type as a shortcut? Four. I'm fine with that. It's just sometimes the math will be yucky enough that you can't do it in your head but yeah I'm going to put a four there. Amy how many numbers do I have on top? More than one? Better put brackets. Okay for me to type this it's going to be Amrit Bracket. Four minus, are there any exponents here? Then I can probably survive without the remaining brackets. Negative three times negative 8748 closed bracket divided by four. And I get negative 6,560. Yes? No? People nodding? By the way, Jim in this equation here there is one two three four, sorry one two three four different variables. If I give you any three I expect to be able to find the missing one and same with this equation here. If I give you any three of those I expect to be able to find the fourth one. Do a bit of equation solving. Example three, turn the page if you haven't already. The sum of a certain number of terms in this series is that. What is the last term that would make this series add up to that? What's this question asking me to find? L. So I'm going to write down L equals question mark, which right away commits me to a certain equation. This one S equals A minus RL all over one minus R. That was from memory. Am I right? Woohoo I am. In physics I'd probably get the L by itself first. You know what? I'm going to be pragmatic and just start sticking in numbers. Because it's going to end up being cross-multiplying I think. Read this question very carefully. What's S the sum? Not a negative 104,0858. It says the sum is that. And then it says we'll make it add up to that. So it adds up to negative 104,858 equals. I guess that means they must have told me A did they? What? Negative two minus. They must have told me or I can figure out R negative four. L is what I'm trying to find all divided by one minus negative four. I'll be honest. You know what I would probably do right now? I would probably with my pencil cross that out. And instead of a one minus minus four what's a nicer thing to write there? Five. I would do that. How can I solve this equation? Please God let them remember. Please God let them remember. Please God let them remember. Matthew please God let them remember. You know what? I think I can cross-multiply. One times that equals five times that. Negative two minus negative four L equals five times that. Five times negative 104,858. Negative five, two, four, two, ninety. By the way, what's a minus minus the same as? How would I get the L by itself? Plus two and then divide by, I think that I can do in one step. Plus two, divide by four. And I'm kind of hoping it works out evenly. Hey, it does. The last term is negative 131072. Is that okay? Oh, it gets better Nikol. For example, four in a geometric sequence, the fifth term is 1024. And the common ratio is four. Find the sum of the first seven terms of this sequence series. Hmm. Well, I think they want me to find S seven. Did they tell me the last term? Did they tell me term seven? Nicole just shook her head. So that means we're going to use the sum equation without the L in it. A bracket one minus R to the M all over one minus R. Do I know R? Check. Four. Do I know N? How many terms they want me to add up? No, nothing. Seven. Do I know A? Megan, you know this question is really asking me to find. Really, it's asking me to find A. What piece of information haven't I used yet? The fifth term is equal to what? 1024. I'm going to leave a space over here. I'm going to draw a little line. And I'm going to write down for a single term, TN equals AR to the M minus one from last day. Term five is equal to A times four to the fourth. Five minus one. But I know what term five is according to this question. What is term five? What's the fifth term? 1024 equals A times four to the fourth. Divided by itself. Divided by what? Four to the fourth, did you say? That's a four to the fourth. R is four. And then N was five. What's five take away one? Four. Four to the fourth. You know what? A is going to be 1024 divided by four to the fourth. You know what the first term was? Four. It says many goals every other swing. Here we go. What's A? Four. Bracket one minus what was R? Also four to the N. Now here N is seven all over one minus four. By the way, instead of typing one minus four in my calculator, what is one minus four in the denominator there? I'm going to type negative three. What do you get? What is the sum of the first seven terms of this particular sequence given the information that they gave me? And yes, this time I'm not doing it on my calculator, so you have to try it yourself because it's important that you figure out how your calculators work. Freeze the screen. That's a good idea. Now I can try this myself. Do you have an answer in the 20,000s? Do you get that? 21,844? Or does Matt would say 2184Y? Sorry. Chief Joe. Is that right? Yes? Okay. I take that. Get the whole top in brackets. And I said there's also a bracket around the one minus four to the seven. And then I did the one minus four in the bottom. I just typed in negative three because that much math I can do in my head. And since there's only one number in the bottom. No luck. Don't clear. Applications. Now I have to be honest. Example five, the physics teacher in me hates this question. I love this question. Unfortunately, the provincial exam loves this question. In fact, this is probably on five out of every six exams. So I'm going to cover it. And in fact, I hate it not enough to not put it on your test. It's going to be on your test. A bouncing ball question. So here's what this one is saying. A ball is dropped from the top of a building 100 meters above a paved road. Each bounce, the ball reaches a vertical height that is three quarters the previous vertical height. Determined the vertical height of the ball after the seventh bounce. And then the total vertical distance. We need to draw a picture. I think the ball looks like this. Bounce, bounce, bounce, bounce. And I think I can figure out the pattern from there. How high is the first height? What are we starting at? 100. How high is the next height? How high is the next height? It's three quarters of 75, which is times by three divided by four. What? Louder? I'm going to stop there because here's what I think. I think this is a sequence where A is 100 and R is, I'm going to go 0.75, three quarters, three over four. Here's one of the many reasons I don't like this question. Dina, you have to be very, very careful in your counting. Dina says determine the vertical height after the seventh bounce. I think that's term seven. I think. But I'm not positive. Let's see. Here's the height after the first bounce. What term is that? Is that the first term? What term after the first bounce? What term is this? After the second bounce, what term? Three. After the seventh bounce, what term am I asked to find? There's the first sneaky bit of wording I don't like. G, would they put term seven as one of the answers on the multiple choice of the provincial? Do you think yes? I think that's cheap. There's better ways to word it in my mind. Anyways, it's going to be A, R to the M minus one. What's A, 100? What's R, 0.75? I'm sure I, I'm sure last lesson I would have done the pirate movie joke, right? I went to go see a pirate movie the other day, but I couldn't get it because it was rated R. Okay. Really? That gets last from Vitali. Really? Okay. What's N, that number? What's N minus one, right? The eighth term, the exponent's going to be a seven. Okay. What's the height after the seventh bounce, which is actually the eighth term? 100 times 0.75 to the seventh. What's the height to the nearest tenth, 13.3? Yeah? B. What's the total vertical distance that the ball travels? And here's why this question is so tough. I have to include the distance when the ball is traveling down, but I also have to include the distance when the ball is traveling up. This is an Adam-up-some question that's total. There's several ways to do this. I think the easiest way to do this, and again, we're going to redraw this quickly. So we have one, two, three, four bounces. I think the easiest way to do this is to add up all the on-the-way downs. That's those sections where this is 100, this is 0.75. We'll use a summation equation, not 0.75, 75, but we'll use a summation equation, and this is 56.25, et cetera. Look up. I think the easiest way to do this is to add up all the yellows times that answer by two, which will give you that, that, that, and here's the problem, it'll also give you that. Oh, minus what? Okay. There's several different variations on how to do this. I think the easiest is add up all your downs times by two minus the original height. Or if I want to write this mathematically, Jen, add up, ooh, I better count. Contacts the floor for the seventh time. First time, first term. Second time, second term. Third time, third term. Do the term and the bounces match this time? Yes, they do. Dina, I want s7 times two minus 100. What's s7? Well, s7 is going to be a bracket one minus r to the m all over one minus r. Madison, what's a? 100. Bracket one minus, what's r? 0.75. What's n? 7 all over one minus 0.75. By the way, on my calculator, will I type one minus 0.75? No. What will I type instead? 0.25. I think I can do that in my head too, yeah? 100 bracket one minus 0.75 to the seventh all over 0.25. Get s7 and all of you, all of you, all of you try this on your calculator right now. I think s7 is 346.6064. Vitaly is holding up his arms outstretched because as far as he's concerned, the Canadian silver chance. At least his calculator came through. Yeah, I've done that yet. Do you understand what the point was though? It was to take into all this, we're nowhere near done. This, this, which is this here times 2 minus 100. So I'm just going to write down this answer first of all which was 346.6 and I'm going to plug that into there times 2 minus 100 because we counted the first height twice in our method. 593.2, get that question because the physics in there bugs me. Well later on we'll add up an infinite number of bounces. That's the one that I hate the most because you know what? It doesn't keep bouncing for infinity in real life. There's energy, anyways. I'll let it die. Which it would because it'd be energy loss and it would come just, well, what if you're bouncing it on a bouncy castle floor? Yeah, okay, fine. Amit, did I answer your question okay? Good, turn the page. Let's see asking me to find how many times but which variable is that? S, A, R, N, something else? Brendan, forget it. There's limits to how much I'll put up with this question. Sometimes what they'll do is they'll give you a summation equation but Evan, Brett, they've tied it up for you. Here's a great example. They're saying look, we've taken the summation equation but we've simplified, we've tied it up and in simplest form it looks like that. It says find the first four terms of the geometric series defined by that. In other words, find S1, find S2, find S3, and then over here find S4. S1 is going to be 5 bracket 3 times to the 1 minus 1. I'm just plugging the N directly into this modified, simplified equation. S2 is going to be 5 bracket 3 squared minus 1. What the heck is that? Well, I can do this first one. What's 3 to the 1? 3 minus 1 times 5. Apparently if you add up the first one terms, you get 10. Actually, I think that also means the first term is 10 because if you're only adding up one term and it's the first term, I guess the first, I guess A is also 10. Little hint, Elizabeth, A is S1 always. Handy back pocket trick. S2, Elizabeth, what's 3 squared? Minus 1 times 3. Sorry, times 5. Yo. And again, Elizabeth, I'm just using this equation here. So let's do S3 together, my child. So it's going to be plugging in N for the N. What's 3 cubed? 27, a little trick here. Minus 1 times 5. 130, I think. Yes. What's S4? That's going to be 5 bracket 3 to the 4th minus 1. 3 to the 4th is 81 minus 1 is 80 times 400. Now, here's what else I know. I can actually, from this data here, I can actually get all four of the terms. I know the first term, if you add up only one term, the first term must have been 10. Now, when I added up the first two terms, what did I get for an answer when I added up the first two terms? 40. So if the first one was 10, what must this one have been? Must have been a 30. There's my 40. Oh, and when I added up the first three terms, I got 130 as an answer. If I had a 10, if I had a 30, what would go here so that I got 130 as an answer? 130. I have 10. I have 30. I already got 40. I want to get a total of 130. I think that must have been a 90. And you know what this number must have been? This is adding up the first one term. This is adding up the first two terms. So if I know the first term is 10 and I know they add to 40, what does the second term have to be? It's got to be 30. This is adding up the first three terms. It's saying this plus this plus whatever was here gave me 130. I got 10. I got 30. That's 40. I want 130 as my total. What number would go there? 90. Actually, there's a better way. Anybody see the shortcut yet? How can I tell you right away that this is 270? Even easier than that. Even easier than that. Look up. Look up. Look up. And stop talking. Look up. If I take any sum, the fourth sum, and I subtract the third sum from it, what is 400? Take away 130. If I subtract the third sum from the second sum, that gives me the third term. It tells me what you added, what extra number you added on. Any sum subtract the one before it gives you the term at that location that you just added on. So it says find T9 without using this. The ninth term is going to be the sum of the first nine terms minus the sum of the first eight terms. It's going to be five bracket three to the ninth minus one. Take away five bracket three to the eighth minus one. It's going to be five bracket three to the ninth minus one. 98,410 minus, instead of a nine, put an eight there. 32,800. That's adding up the first eight terms. That's adding up the first nine terms. The difference between them should be the ninth term itself that you just added. Nine, eight, four, one, zero minus three, two, eight, zero, zero equals 65, six, ten. There's the ninth term. That's a handy to have in your back pocket gem that if they ever give you sums and they say find a term, any term is the sum of that number subtract the sum of everything one less than it. Can you survive without it? It shows up every so often, but probably you can survive without it. So part C here says, find an expression for TN by two different methods. Well, the first one is what we learned last day or add up the first N terms. Take away, add up the first N minus one terms. This one here, not on your formula sheet. So there it is. I won't put a big highlighter around it because sadly it's not on your formula sheet but it does come in useful. What is your homework? How about one A, E, and H will go diagonally. Number three, number four. The first term is three, they give you A. Some of the first two terms, some of the first three. You're going to die? We all die, my son. 12 and 13. I didn't give you much homework on purpose because you got to test next class. Your other homework is work on the probability review assignment, which is do next class. That's the one that looks an awful lot like this bad boy here. There is a handwritten answer key online for this one here. You have the remainder of class.