 We're going to go way back. Turn to page 163 in your workbook, please. And yeah, we're using the workbook again because it does a much better job. The other reason I use those photocopied notes, by the way, is they do combinatorics and probability in one less lesson than the workbook, so we gain two days, which is more handy than you realize, all right? And most of my students have told me this final mini unit is, I hesitate to use the word easy, but it's kind of nerdily interesting and not too bad. Certainly, I don't think it's on the same level as trigger logs. Sequences is what we're going to be looking at. So we're all on the correct page that I've said twice now, but I'll say it one more time. Page 163, page 163 under the B1, under the I6, under the M3, thank you, Elizabeth. Nice try trying to not look down and turn the page so no one would notice, but I caught that. Not as smooth as you thought. And now it's preserved for posterity on the internet. Now that I'm on YouTube, by the way, I'm getting people from all over North America subscribing to me, it seems, because I don't recognize any of the usernames. So you may be international. Woo-hoo! A sequence. A sequence is a number or a set of numbers in a definite order which are separated by commas. It's a group of numbers that have a pattern. The terms of a sequence are the values of some function whose domain is the set of natural numbers. That's a fancy way of saying, Jen, one of the things we're going to try and do is come up when we see a pattern of numbers with an equation that will generate any term without us having to write all the ones in front of it. So we're going to try and spot the pattern for the first four numbers. And I'm going to say, what's the 80th number? And please don't write the first 79. Come up with an equation. OK? A common way to define this function is by a formula which gives the nth term the generic term of a sequence. So let's start out. It says, write the next two terms of the following sequence and then describe a rule. Very basic. You've done this, hopefully, since you were children. Spot the pattern. What comes next? 9 comma 11. Now here is our terminology and our symbols. We call this first term, and I'm going to zoom in so I can write really small. We call this first term, term one. We call this second term, term two. So what would this guy be called? Term number what? Five or six? Which one? The 11. OK. This is term six. What I want to do is I want to write an equation so that when I put a six into the equation, the answer is 11. When I put a two into the equation, the answer is three. When I put a one into the equation, the answer is one. Sorry. OK. So here is Evan's suggestion. The nth term, the fifth term, the sixth term, the tenth term, the nth term is equal to 2n plus one. Now does that work? Let's see. If n is two, if I put a two there, I should get a three. Ah, 2n minus one. Does that work? Put a six in. Do you get an 11? Yes? Put a five in. Do you get a nine? So once I've done that now, Dina, I could say, hey, what's the 43rd term, and you don't need to write out the first 43 or 42, OK? This is the generic formula. By the way, Brendan, it's two times 43 minus one. It's your two times table. You're really reaching for your calculator? Really? It's your two times table. Come on. B. What comes next? 32, 64. So this is term one, term two, term three, term four, term five. This is term six. Is there an equation that if I put a six in, I get 64 back. If I put a five in, I get 32 back. Evan, you're on a roll, what? The nth term is two to the nth. Let's see. Two to the one is two. Two squared is four. Yeah, that works. What about C? What's the next term? 250. What's the next term? 125, OK. This is term one, term two, term three, term four, here's term five, here's term six. Can anybody come up with an equation so that when I put a six in it, I get 125 as an answer. When I put a five in it, I get a 250 as an answer. When I put a one in it, I get a 4,000 as an answer. You ready? This is the type of sequence we're going to look at this unit. It has a special name. It's called a geometric sequence. And it's a sequence where you're multiplying each term by the same number over and over. Now I know I'm dividing by two, but I want to talk about that in terms of multiplying. What am I multiplying each term by? What's the same as dividing by two? A half. This is a geometric sequence with a ratio of a half. And here is how you write the equation. Just watch first without writing down, see if you can figure out where I'm getting these numbers. Where does the 4,000 appear in the original sequence? First term. Where does a one-half appear in the original sequence? It's the common ratio, it's what I'm multiplying by. And then I added the n minus one because I've seen these before, but watch, Elizabeth. If you put a one right there, what's one take away one? Zero. What's anything to the zero power? A half? No. What's anything to the zero power? Ah. Putting a one in does give me a 4,000. It gives me 4,000 times one. Dillon, that's why I had to do an n minus one. If I want to put the first term there, put an n minus one there that gives me my first term. Let's try it for this one. Put a two there. Let's two take away one. One. What's one-half to the one? What's half of 4,000? Works. Ooh. Let's try a six. 4,000 times one-half, I'm going to type .5 to the what's six take away one. Will a six give me 125? Oh, I can hardly wait. Let's find out. Yes! We're going to come back to this special sequence here. A couple more just for practice because we're nerds. D, what's next? 13? What's next? 17? What's the pattern? By the way, if you could come up with an equation for that, you'll win the Nobel Prize for math. It hasn't been found yet. And we're interested because right now, all of your internet encryption, if you ever buy something online, your Visa card is encrypted, and the way it's encrypted is based on multiplying two huge prime numbers together. Prime numbers that are about 50 digits long because multiplying is much easier than going backwards and dividing. That result is used as what's called an RSA, public key encryption to encrypt your stuff. So if you did find a formula for that, although you would win the Nobel Prize for mathematics, it's called the Field Medal, by the way, you'd probably bankrupt the internet economy because now you wouldn't be able to encrypt credit cards. E, 8, 13, what's next? 21, what's next? 34, what are we doing? How do you find each term? Okay, so here's how I would write that. The nth term is equal to take the term before it and add the term to before it. I think that's how I'd write it. This is called a recursive sequence because it has T's on both sides. We're not looking at those. It's also called the Fibonacci sequence. If you've watched the Da Vinci Code or read the book, it's popular in there. Sadly, I knew of it long before that. So as soon as I saw those numbers in the book, I'd solved it already. It wasn't quite as exciting for me as it was for some people, but still. It shows up in weird places, the Fibonacci sequence, all over the place in nature. So the nth term, the nth term, the nth term, the nth term, we call that the general term of a sequence. It's the generic equation to generate a sequence. So the first term is written as T1, second term as term two. The two is subscripted, so you don't think you're multiplying. The general term or the nth term is written as Tm. The general term provides a formula which is used to describe any term of a sequence. For example, the first one Evan pointed out was the nth term is 2n-1. We break sequences first of all into two main types. Finite and infinite. Infinite has a dot, dot, dot, and that goes on forever. Finite often has a dot, dot, dot in the middle, but then they add a last term in what they're saying. Their DNA is, stop there! By the way, what's the first term in each of these sequences? Three, what are you multiplying by? Two, can you see the general term? It's first term times the ratio to the n-1. You're going to eventually not have to memorize that. It's on your formula sheet, but it's just really here. That works for any sequence that's generated by multiplying the same number over and over and over and breadth. Those are the ones we're going to spend some time on. Next page. Example A says, list the first three terms of this sequence. So, term 1, term 2, term 3. What this really means, Brett, is put a 1 in for n. So, what's 1 squared? Times 3, 3 minus 1. The first term of this sequence is 2, gamma. Madison, what's n for this one? 2, 2 squared. Times 3, minus 2. I guess the second term of this sequence is 10. Try finding the third term on your own. Your hint is it's 3 squared times 3 minus 3. 3 squared is 9. Is it that? B says, find the eighth term. How would I find the eighth term? Put an 8 in for n. n is the term number. I'm not going to bother. I said we break sequences into two types, infinite and finite. There's another way we break sequences. We divide sequences into how they're created. There are what are called arithmetic sequences, geometric, and others. Arithmetic, I think you did some in math temp where you're adding the same thing over and over. I'm not going to look at those. Well, it says the constant is called the common difference. What are you adding in each term here? Let's at least write that down. The common difference is 4. We're going to look at geometric sequences where you're multiplying the same thing over and over and over. We're going to say multiplying. If you're dividing by 3, we're not going to say that. We're going to say you're multiplying by 1 third. That way it lets us always write the same style of equation. We call what you're multiplying by the common ratio. What's the common ratio in this example here? How can you find the common ratio? Most of you are doing it intuitively, but if it's yucky, divide any term by the one in front of it and give you the common ratio. 6 divided by 2, or 18 divided by 6, or 15 divided by 18, either of those, pick whichever one's easiest. That's if it's a yucky one with decimals and you can't see the common ratio. Any term divided by the one in front of it. Then there's other sequences. We're not going to care about those. Example 3 says classify the following as arithmetic, geometric, or neither. I'm just going to circle the geometric ones. This one here, you're multiplying by 2. This one here, you're multiplying by a half. Those are the two geometric ones. So geometric sequences that we're going to focus on for the next four lessons. First thing you've got to be able to do. Amateur, you've got to be able to find the ratio. And often you'll just do it intuitively. What are we multiplying each number by here? I'm willing to bet not too many of you went 12 divided by 6. You just somehow saw it. That's the best way. R equals 2. But if you're not sure, any term divided by the one in front of it. Now this is not on your formula sheet so I won't do a big highlighted thing around it. But I'll be repeating that over and over. So what's the common ratio here? What's the common ratio here? Don't squint any term divided by the one in front of it. Dina, any term divided by the one in front of it. Now I'll simplify that. Any term divided by the one right in front of it. See how I got that? But what is negative divided by negative? What's 5 over 10 in lowest terms? In other words, if you can't just do it intuitively like we did here, any term divided by the one in front of it and pick the easiest two terms, I could also have gone negative 5 halves divided by negative 5. Why deal with fractions if you don't have to? Here's one they love to do. Put a little D before you turn the page. I like this question. I like this question. I like this question. Y squared over X, Y, X over Y. What's the common ratio? This is how they get around the fact that there are apps that you can download for the calculators that will do a lot of this for you. How can I always find the common ratio? Any term divided by one in front of it. I think I'm going to go this one divided by this one. Y divided by Y squared over X. You guys are all looking panicky. Oh, come on. Why don't I do that there? Did you say cross-multiply? Am I getting my baseball bat out? Am I going to have to freak on you? That's not cross-multiplying, Evan. Cross-multiplying is when you have one equation equals one. How do I divide by a fraction? You have never heard me say cross-multiply. You know what? I won't use a time sign because it's going to look like an X. I think the common ratio is that. Is it not? And I think you also get the same thing if you do it this way. Or do you? X divided by... Did I do this right? Oh, you know what? I should have done this. Erase the Y in the last term. That's better, I think. Negative 1X is 0X is 1X. 2Y is 1Y is 0. Yeah, there's the sequence. Sorry. Then that would have been a way smarter thing to do. Any term divided by the one in front of it. What's X divided by Y? It's that. That would have been an easier way to find the ratio. But they love to give you algebraic ones. They just don't like to make them up on the fly like I did because apparently it's too easy to make a dumbest rake. Sorry. Turn the page now. It says developing a formula to find the general term or towards the end of the year. If this was in October, I would actually develop the formula with you. Instead, I'm just going to go straight to the punchline. We're going to scroll down, down, down, down, down. Here's what we're going to say. This here is the formula for any geometric sequence. But we need to talk about what each letter means. Some of these letters, you already know what they mean. What does R mean? Ratio. What does N mean? The term number. Term 5. Term 7. Term 19. 342. And A is what we chose to use to represent the first term. Why do you think we picked the letter A to represent the first term? Yep. It's one that makes sense. That's all right here, by the way, but I've always found kids by writing it out, it jogs in their memory a bit more. Okay? So look up. How would I write this as a general equation? The nth term is equal to, Nick, what's the first term? A bracket. What's R? More specific in this. What's R to the n minus 1? Does that work? Let's see. The fourth term is a 24. If I put a 4 right there, what's 4 take away 1? 3. What's 2 cubed? 8 times 3. 24 works. Here you want to come up with the equation. The nth term is going to be, Amy, what's the first term? What's R? You know what I would do? I would use these two here because they're smaller numbers. Any term divided by the one in front of it. First of all, is R going to be negative or positive? Negative. And in lowest terms, kiddo? How about a half? Yep. 4.5. Either is acceptable to the n minus 1. So do you have to memorize this formula? No. It's on your formula sheet. You do got to know what the letters mean though. But I think they're picked fairly intelligently this time. In fact, you can think about a sequence this way. Madison, give me a number between 1 and 10. 7. There's the first number. 7. Dylan, give me a number between 1 and 4, including one or, but not, sorry, between 2 and 4 inclusive. Let's let R be 3. So 7, 21 times 3, 63 times 3, 189. If they gave me that, algebraically, algebraically with tally, I think of it as term 1, term 2, term 3, term 4. I think of it as A, first term, A times R, because if N is 2, R to the n minus 1 is just R. AR squared, AR cubed. Think of it that way. Next page. So you're ready? Example 5, do both of these on your own. Find the general term and then find term 7. It says, determine the general term and calculate the seventh term. And yes, this time you can use your calculators. I'll freeze the screen, see if you end up where I end up. There's the answer to A. See if you get the same thing. And by the way, it's customary to use fractions rather than decimals as your final answers. So the .008, whatever it was that you may get. Nope. Is it negative or positive? I'm sure. No, I think it's positive. Final answer is positive. 1 over 192, yeah? Skip 6. Example 7. Now, what I can also do is instead of saying here's the first couple. Find the general term. I can tell you a specific term at a location and find either A or R or M. So here's what I would like us to do. Write above example 7 where you have some blank space. Let's write down our generic equation, which is TN equals A, R to the N minus 1. That's what we put in the little box of the previous section. So it says, given this sequence, determine what's part A wanting me to find in this particular equation, which variable? David, N. That means they told me everything else. Let's see. Let's put an equal sign. Did they tell me A, the first term? What's that? Good. Did they tell me R, the common ratio? 2. You've already said I don't know Ns. I'll put the N minus 1 there. Did they tell me the Nth term? Did they tell me the term at the location that they want me to figure out? What? 16,384. Where is the variable sitting in an exponent? How do I solve an equation where the variable is in an exponent? Logs, baby! Oh, but wait a minute. We're not going to take the logs yet. We always want it to get the exponent by itself. I see right under there. Let's divide by 32 first. Divide by 32. Divide by 32. Our new equation is going to be 16,384. Divide it by 32. 512. I have 512 equals 2 to the N minus 1. Where is the N sitting as an exponent? Log both sides. The log of 512 equals the log of N minus 1. Why does the log help? What can I do with that exponent? Oh, pray tell. Isn't this great review? Where's the 2? I'll make it a little bigger so you can see. Sorry. And I think I actually forgot to write it. Sometimes my tablet's acting up. Sometimes I write stuff but it doesn't show up. You guys have occasionally seen me going, nothing's appearing for you. I don't think this time that was the case. What can I do with that exponent though, Madison? Move it to the front. In fact, I'll get this. The log of 512 equals N minus 1 in brackets because there's two terms. Log 2. Is that how to get the N by itself? How to get the N by itself? I think first I would divide by log 2 and then plus 1. In fact, N is going to be the log of 512 divided by the log of 2 plus 1. And it's been so long since you touched that log button. Many of you, why don't you all practice piping that in just so you can get familiar again with the procedure, right? Oh, by the way, N, because it's a term location, can't ever be a decimal. If you get a decimal, you know you messed up. It's not the third, the 3.5th term. That's the third term or the fourth term. What do you get? 10? B. So B is the same question as A. It's saying which term, really, which term number is 1024. So try this, but instead of a 16384, they're telling you the term is 1024 and see if you can solve for N on your own. It's a great review of logarithms. I'll do it slowly up here with the screen frozen. So, if I haven't said so already, I like that question. I like that question. I like that question. That's a great multiple-choice question. Find N. Oh, I'll even tell you what the answers would be for question B. Six, seven, or something else, silly. With the N minus one, the most common mistake, kids find N and it's one too big because they forgot the minus one. Example eight. I like this question. I like this question. I like this question. I like this question. I like this question. Absolutely going to be on your question. Now we're going one step further. Now they're telling me the fourth term, what's the fourth term of this sequence? And the seventh term, what's the seventh term of this sequence? Find the first term, the ratio, and the general term. We're actually going to find the ratio first. There are several ways to do this. Nick, there's a nerdy algebraic way and although my heart likes it, my head does not because I think there's an easier way and the easier way is to draw this. Put your pencils down and look up. What term number is this here? I don't know term one. I don't know term two. I don't know term three, term four. I don't know term five. I don't know term six, term seven. Write that down. It takes five seconds to draw that and it gives you a nice visual representation. And we're going to use this to get an equation. Once you've written that down, put your bets down because this is where we're going to get clever. Remember I said, we're going to find r first actually even though they said find the first term. I always find r first. So what did I multiply this number by to get that? I don't know but I know what I called it. What variable have I used for the thing that I multiply by? And you know what I multiply this number by? And you know what I multiply this number by? See how I drew the comma there? Because here's what Sav is really saying although he doesn't realize it yet. He's saying, hey, Duke, if you start with a negative 54 and you multiply by r three times or r cubed for short, you end up at, there's my equation, and I think, did you say an easy to solve? Did I hear you say easy to solve? Yeah, how? Divide by negative 54 and then how do I get rid of a cubed? I like this method. There are other methods and systems that cross-multipline. I think the one second it takes to draw it, it falls apart so nicely. Does that make sense? Okay, Brett. Good. Find r. Oh, this is also a good chance for you to review where your cube root and fourth root and fifth root buttons are on your calculator because you're going to be using this great review. r cubed is going to be 1458 over negative 54. r is going to be the cube root of 1458 divided by negative 54. Now, technically, I know the answer to this without a calculator. I'm pretty sure it's negative three, but you should know where your cube root button is, so find the cube root and negative 27. Confirm for me that it is negative three. It is, yes? So, Megan, what if I'd given you this same statement? There's term four. There's terms instead of five, you know? There's the seventh, eight, nine. You could just hop Scott's your way. Or find the third term. You could going this way, you would divide. Oh, they want me to find the first term. Ready, Seth? What did I divide by if I'm going this way? What did I divide by if I'm going this way? What did I divide by if I'm going this way? It seems to me if I start with negative 54 and I divide by one, two, that many, I end up at the first term. Oh, and do I know what R is? That's why I said I always find R first. I think A is going to be negative 54 divided by negative three. Oh, no, negative three cubed or cubed. What's A? Two or negative two? Two. And you know what I tend to do just because I feel so proud that Tally, I tend to say, oh, not a T, Mr. DeWitt, a two. I know the first term. What's the last thing want me to find, the Tally? What's the third thing? The general term. Now the general term equation, which you don't have to memorize, it's on your sheet was the nth term is equal to A, R to the n minus one. So let's plug stuff in specifically for this particular sequence. The nth term is equal to what is A? We figured it out. What? Two. What's R? Negative three to the n minus one. There is the generic term to generate that sequence. So if we're, you know what I'm even going to say, Jen, if we're only medium clever, we can find whatever the heck they want to if they give us two terms. Am I ever going to give you like the third term and the 80th term? No, I'm not going to make you draw that much. It's going to be terms that are three or four or five apart. You'll end up doing a cube root of four through the fifth, through the sixth. That's my method. In the workbook, who has the solution manuals? He will use the algebraic cross multiply method if you don't like my method, use that one. It's so much fun. Turn the page. I like number nine. I like number nine. Number nine is a nice question. I like number nine. I like number nine. I like number nine. And here's what it says. X plus five and X plus nine are the first three terms in a geometric sequence, a sequence where you're multiplying by the same thing over and over. Find the exact value of each term. I think really what I'm going to do is find what X is and then just put it in for each of those. Hmm. Well, Sam, what did I multiply by to get from this term to this term? Algebraically, what variable have I got? What variable have I used? What did I multiply by to get from this term to this term? Algebraically, what variable have I used? So here's what I think that means. Put your pencils down and look up because I'm going to do kind of a three-step conclusion and then wrap it up. It seems to me how can we find R if it's numbers? What did I say always works if you can't spot it? Any term, what? Divided by the term in front. It seems to me that X plus 5 divided by X equals R. It has to. Oh, and it seems to me that X plus 9 divided by X plus 5, any term, divided by the one in front of it. You know what that has to work out to? R. If those both equal R, what can we conclude? They equal each other. So I'm going to get the equation like this. X plus 5 over X has to equal X plus 9 over X plus 5. Any term, divided by the one in front of it has to equal any other term divided by the one in front of it. Evan, how will I solve this equation, oh, pray tell? What's this a wonderful example of? This is cross-multiplying. You got it, baby. I'm going to rewrite this as X plus 5, X plus 5. Oh, this is going to be a quadratic. It will be nearly cool. X times X plus 9. Now what? Freeze up and lose marks? No, no, no. You've done the hard part. Now what? Factoring is when you write it with brackets. We want to do the opposite of factoring. Our acronym for that is usually FOIL. Get rid of brackets. So the right-hand side is easier. X squared plus 9X. The left-hand side, X plus, oh, this is FOIL. I think X squared plus 10X plus 25. What kind of an equation is this? Now, I thought it was going to be a quadratic. Actually, it's not, because what's going to happen to the X-squareds when I try getting them to the same side? When I minus X squared from both sides, it's going to vanish. I thought I might have two values for X and two possible Rs that worked. I like that question. I like that question. But here, I think Dina, cancel. Cancel. In fact, Jen, I have 10X plus 25 equals 9X. I'm going to minus 10X from both sides. I get 25 equals negative X. I think X equals negative 25. So if they want the first three terms, and it's based on this pattern, what's X? What's the first term, also X? Negative 25. What's the second term? Negative 20. Negative 25 plus 5. What's the third term? Negative 16. What's R? If I ask that instead. Any term divided by the one in front of it, so negative 16 over negative 20, positive, reduce the fraction. What's the seventh term? Now that I know A and R, could I find term 7? Yeah. I mean, whatever they need. Sorry. But this algebraic one where there's three terms with X's in each of them and you don't know X. I like that because it's reminding you you do know R. It's any term divided by the one in front of it. And any pair of Rs has to be equal to each other. One more. Do you know what I mean by that? I can't call the math easy exactly, but you've got to learn some stuff. Also though, you've seen a fairly nice review of some logarithms. A geometric mean. Does anybody know what does the word mean in math? It's a fancy word for average. So, a geometric mean is a fancy term for the terms being put between a geometric sequence. And it's harder to explain than to show. Jen, can you read example 10 to me please? Here's what this means. Starting with 81, and ending at 129, you know how many numbers they want us to put between them? Four. They want us, Jen, to do one, two, three, four and one over 729. That's inserting four geometric means. What if they said three geometric means three numbers? Saf, does this look kind of similar to this one here? So, Saf, my friend, I ask you, hey, what did I multiply this by to get to there? I need to come up with an equation that has a little d in the middle so it could be r, d, r, r, d, r, d. Saf, my friend, how many r's are there to get to 1 over 729? Can you tell me what equation I'm going to leap to? Starting with an 81, r to the fifth equals, come on, 7, there we go, 729. How would I get the r to the fifth by itself? r to the fifth equals 1 over 729 divided by 81. Math, enter, enter. Oh, it won't do this one as a fraction for me. Now, I still can do this as a fraction. When I divide by 81, isn't that just like putting an extra 81 at the bottom? The denominator of this fraction is going to be 729 times 81. It's going to be 1 over 59049. That's r as a fraction. You okay how I got that, Jim? Dividing is same as just multiplying it onto the denominator. How do I get rid of a fifth? Fifth root? r equals the fifth root of 1 over 59049. Now, I wonder if it'll work with the decimal let's see. Fifth root, oh, where's fifth root? Math. Oh, I got to type 5 first. Fifth root. Math, option 5 of that answer. Ah, you know what r is? 1 9th. So, you ready? Let's put the numbers in between. How would I find the first one? 81 times 1 9th. 9. And then take this answer times 1 9th. 1 times 1 9th. I think that's 1 9th, is it not? And I think the last one is 1 over 81. I could also have just divided by 9 each time along the way. There they are. Geometric sequences where you're multiplying by a common ratio. First term a, r is the ratio any term divided by the one in front of it, Amy. And n is the term number, the number of terms. Tn is the actual term of value. What's your homework? How do I get the r to the 5th by itself? So, dividing by 81 is the same as putting an extra 81 down there. If you go 81 times 7 29, that's what that is. That, as the decimal, but if this was multiple choice, the fraction would be the answer to Pickrum. Now, of course, you'd be clever enough to take those fractions and change the decimals and see which one matched your decimal, but still. Is that okay? Brett. Number 2. Number 3. So 2 and 3. 5, 2, 3, 5, 6. Take a look at b. What is the last sentence of b? Explain what how much you want to bet this time you do get a quadratic where there's two x-values that work. Unlike the quadratic where we had that canceled out. 8, 9, 10, and 11. Everybody do this. That's how many more lessons. Because 4, take away 1 is 3. We are in the home stretch.