Common Core Algebra I.Unit 4.Lesson 13.Arithmetic Sequences
Algebra 1
Hello and welcome to another E math instruction common core algebra one lesson. My name is Kirk weiler. And today we're going to be doing unit four lesson number 13 on arithmetic sequences. This is a follow-up to the last lesson on sequences and I'm going to assume that you've actually had an introductory lesson on sequences before we get into arithmetic ones. All right, but let's dive right into it. Now, I'd like to remind you before we get started that if you want to copy of the worksheet used with this lesson or an accompanying homework, you can click on the video's description or visit our website at WWW dot E math instruction dot com. As well, don't forget that on the top of every one of our worksheets, especially next year when the entire workbook is published, there are going to be QR codes. Those QR codes allow you to take a smartphone or a tablet and come directly to this video. All right, let's get into the first exercise. Let's motivate the discussion by talking about real world example. Let's say we have Evan. Evan is saving money to buy a new toy. She already has $12 in her account. She gets an allowance of $4 per week and plans to save $3 in her account. Fill out the table below for the amount of money Evan has after N weeks of savings. So remember, she's going to save $3 per week. All right? So after one week of saving, she has $15 because she started with 12 in her account, and she's going to add in $3 per week, so she starts after one week with $15. Then adds in another three to get to 18. And she's going to add another three and get to 21. Then she's going to add another three and get to 24. Et cetera, right? All right. Now, this is textbook definition of a sequence, right? It's just a list of numbers. It's 15 comma 18 comma 21 comma 24, et cetera. And keep in mind what we did is we started with 15, and then we added three to each one. Now letter B asks us to write a recursive definition for the sequence. Recursive definitions are ones where you give the first member of the list or maybe the first couple members of the list. So a one, the first thing in the list. That just tells us where we are in line. The first one in line is 15. Then what we do is we talk about the n-th one in line. That's an N and we say, well, the n-th one in line is going to be the one right before it, which would be a sub N minus one. Plus three. So remember, these things which are known as the index. Are where you are in line. Okay? So when we write something out and when we interpret something like this, we interpret it as wherever you're standing in line, you figure out that number by looking at the one right before it, and adding three. All right, take a look at letter C, it says what's wrong with the graph of the sequence shown below. And let me just give you a hint. I plotted all the points right. But there's something fundamentally wrong with the graph. What is it? Pause the video right now and think about this. All right, this is actually quite important. The points shouldn't be connected. So the points should not very important should not be connected. The reason that's so important is that the way that it's drawn right now implies that, you know, after like one and a half weeks of savings, she had $17, but she didn't. Actually, after one and a half weeks of saving, she I'm sorry, wouldn't have even been 17, I think it would have been like 17 and a half. But no, no, 16 and a half. Sorry. Apparently I'm having some problems with math this morning. But she at one and a half weeks, she still only has $15. She doesn't have. 16 and a half dollars. So we shouldn't connect these with a straight line or any type of curve because it implies that we're getting these intermediate values. So it should just be the points. Finally, the problem says Evan proposes the following explicit formula as opposed to a recursive formula. For the amount a as a function of the number of weeks saved N is this formula correct. Test it. Well, it seems like it's good, right? You starting off with like 15 and you kind of got a slope of three going on there. So what's going on here? How about this? Pause the video and test the formula to see if it's right. All right. Well, it should have taken you a very small amount of time to just say, well, all right, let me just test it when N is one. Well, that's going to be three times one. Plus 15 gives me three plus 15, and that gives me 18. But a sub one should have been 15. So that is a big fat no, that formulas not right. All right, we'll develop a formula later on. But that formula is not right. Now, in this really does give us a very specific example of what's called an arithmetic sequence. I'd like you to pause the video now, write down anything you need to, and then we'll go on and we'll talk about arithmetic sequences and give them mathematical definition and all that stuff. All right, let's do it. It's cleared. Okay. Arithmetic sequences are one where each element in the sequence, so each number in the list is a set amount more than the previous number. It's recursive definition is this. I give you the beginning number, right? This is the first number. That's all that one is. Where we are in line. That's the first number. So I'm going to give you that. And then every number in the list is the number before it. Think about consecutive integers. Every number in the list is the number before it, plus some what's called common difference. They call it a common difference because if I have something like this, one comma 7 comma 13 comma 19 comm et cetera, if I do 7 minus one, which is a difference, I get 6. If I do 13 -7, I get 6 if I do 19 -13, I get 6. It's a common difference. But it's also what you add on. To get to the next number. All right. Let me give you an example of this. Let's say I gave you an initial number of two, and then I said every number in the list was gotten by taking the previous number and adding four. Well, this is what it would look like, right? We would start with two. All right, now we would add four, and we'd get up to 6. And we'd add four. Let me get up to ten. At four, get up to 14. At four, get up to 18. At four, get up to 22. Et cetera. We could keep doing this. But that might be it, you know? Now, one of the hardest things about this terminology is that you actually have to know it. It's an arithmetic sequence our sequence that just increases by a constant additive amount. So we're going to dive into some more exercises. I'm going to clear this out though. Not much there to really look at. And let's go on to the next page. All right. Number two, an arithmetic sequence is given using the recursive definition blah. I'm not going to read it off for you. You read it. Which of the following is the value of B sub four shows the show the work that leads to your answer. How about this? I'm going to give you a chance to pause the video right now because I bet many of you, many of you can do this problem already. So pause the video and see if you can figure out what B four is. All right, let's go through it. Now let's just make sure we understand what's going on. Right? What this thing really says, right? So I've got the list of elements that we're calling them B whatever. I've got B one, B two, B three, and B four. B one is negative three. The recursive definition then says I'm going to create each one of these elements by taking the one prior to it and adding 6. So I'm going to add 6, and that's going to give me three. And then I'm going to add 6, and that's going to give me 9. And then I'm going to add 6, and that's going to give me 15. Choice four. That's it. It's not that hard, right? Arithmetic sequences are very important. Probably because they're one of the simplest type of sequence that we can think about. Later on in the course, we're going to look at what are known as geometric sequences. Those are a little bit trickier, but pretty similar. All right, I'm going to clear out the text. You pause the video if you need to. All right, here it goes. Let's take a look at another problem. Exercise three, for each of the following sequences, determine if it is arithmetic based on the information given. If it is arithmetic, fill in the missing blank, if it's not, show why. All right, let's take a look. These are kind of fun problems. Now remember, with an arithmetic sequence, what's got to happen is we've got to either be adding or if you want to think about it subtracting the same amount each time. So we look at letter a and we say, well, okay, what would I have to add to 5 to get 9? Well, I'd have to add four. Now, if you want, you can also think about it as 9 -5 is four. What do I have to add to 9 to get 13? Well, that's also for, right? 13 -9, this four. So so far it looks good. We should check the last pair. What do I have to add to 21 to get 25? That's also for instance 25 -21. That's four. So let's just add four more. And we'll get 17 just to do a little double check 17 plus four is 21. So great. That's an arithmetic sequence. All right, in 17s, the missing number. Let's take a look at B in order to go from 5 to ten. We're going to have to add 5 right again. That's the same as just thinking about ten -5 and getting 5. To go from ten to 20, we have to actually add ten. 20 minus ten is ten. As soon as those are different, this is not arithmetic. We don't even have to check anything else. You know what I mean? Not arithmetic. Because you've got to be adding the same amount each time, and we're not. All right? Let's take a look at this one. This one's a little bit tricky. What do we add to 7 to get negative four? Well, we add a negative start to get positive four. We add a negative three, right? If it has four -7, negative three. To go from four to one, we also have to add negative three. Again, think of it as one minus four is equal to negative three. If you might be saying to me right now, why can't I just think about it as subtracting three each time? That's okay. You can do it that way. Arithmetic sequences work with both addition and subtraction. So that's completely all right. Now let's just make sure this may be a little bit of a trickier one, but 5 negative 5 plus negative three is negative 8. So that looks good too. So let's think about this. What would happen if we added negative three to one? Or subtracted three from one. We would get negative two. That's our missing number. And again, another double check. If I did negative two plus negative three, I'd get negative 5. Awesome. Let's look at this last sequence. Oh. That's 16 -64. That's going to be negative 32. Oh, nope. That's not going to be negative 32. Get rid of my eraser. I actually going to be negative 48. My apologies. All right, let's do four -16. So see how I keep pairing up these differences. That gives me negative 12. Since these are not the same, this is not arithmetic. Not arithmetic. All right, they've got to be the same differences. That's why it's called a common difference. Okay, I'm going to be clear enough the text in a moment, copy down what you need to. All right, here we go. Move it on. All right. I like this one. Exercise four. Consider an arithmetic sequence whose first three terms are given by four, 14 and 24. All right. Now, right away, right away, right? Arithmetic sequences are all about two things. They're about the number you start with, and they're about how much you're adding each time. Should be pretty obvious, even though it's close to the top of the screen. That we're adding ten each time. All right? So then it asks us, what's the fourth term? All right, let's kind of write it out like this. Now I know we know the first three terms, I get it. For 14, 24. Now obviously the next term is going to be 34. But here's the thing that's very important. The thing that's very critical right now is how many times we add ten. Because it would be very easy, of course, to think that for the fourth term, you add ten, four times. But you don't. You only add ten, three times, because you never add it to get to the first term. Think about that. I really want you to understand that. It's very important as we go forth. Right? Because if I wanted to know what the 22nd term was, it would be a pain to write all of them out. So think about B it says use what you learned in part a to find the value of a sub ten, the tenth term. If you think you know what's going on right now, pause the video, then take a shot of this problem. All right. Well, here's the key, right? We are going to add ten, 9 times. 9 times, right? So in fact, we can say that a sub ten will be the value we start with four plus 9 more tens. We're going to add ten, 9 times. That's going to give me four plus 90. And that's going to give me 94. You know, we could certainly do that. We could do that by hand, but it would take a while. And certainly if I told you to figure out the 500th term, that would be pretty obnoxious, you know, to have to do it by hand. On the other hand, just knowing that we added 499 tens, isn't so bad. Okay, let's write a recursive formula. All right. Recursive formula, remember, for that, what we do is we give the first term. And then we tell how the n-th term depends on the one that came right before it. So a sub N is a sub N minus one plus an additional ten. Both of those two elements are very necessary in a recursive formula. The number one mistake students make when they do a recursive formula is that they forget this. So let's say don't forget. I've always told my students that if you forget that initial term, then there would be no way to actually write out the list. I mean, it's great and all if you know that the number that you want in line is the previous one plus ten. But you can never figure out the previous one if you don't know the one before it. You can't figure out that one if you don't know the one before it. And ultimately you need to know the first term. Now, right and explicit formula for a sub N explicit formula. This one's cool, right? So we know we start with the number four. And then what we do is we add N minus one tens. Right? It's what we did up here, right for a sub ten, we added 9 tens. For up here, a sub four, we would have added three tens. And that's it. That's our formula. I mean, don't get me wrong, we could play around with it. We could do things like this. We could take the ten and we could distribute it. Whoops and apparently we could lose an N in the process. Let's try that again. Apparently, I need some more coffee this morning. Maybe we could rearrange. And that's a good formula too. There's absolutely nothing wrong with the one that's in the parentheses Formula One, if you will. But you never know, in a multiple choice context, they might give you something like that. And you have to be able to figure it out. All right, that was a pretty full problem. Now, again, I can't emphasize how important this whole N minus one thing is, right? And by that, I mean, the idea that if I wanted to know, let's say in this problem, what the 200th term was. It would be very easy. Because I just take my four, and I add to it a 199 tens, not 210s, a 199 tens. And just for the sake of argument, I'd get 1994. It was a great year. I think Nirvana was a big back then. Anyway, I'm going to clear out the text so pause the video if you need to. Probably have half the students right now going to Nirvana, who's Nirvana. Okay. Anyway, I've dated myself. I'm going to clear out this text, pause the video if you need to. Here it goes. All right. Let's take a look at the last problem. This one is so cool. Seats in a small amphitheater follow a pattern where each row has a set number of seats more than the last row. If the first row has 6 seats and the fourth row has 18, how many seats does the last row, which is the 20th cabinet. Show your work to justify your response. What I'd like you to do is pause the video now, spend as much time as you need to. I think it could take you upwards of ten minutes to really think about this problem. But see if you can figure it out, okay? Play with it, play with numbers, just play with the problem, okay? But pause the video now. All right. Welcome to go through the way that I think about it. I like pictures. I think they're very, very helpful. So that first row, what did it have? It had 6 seats. I don't know how much the second row has. I don't know how much the third row has. But by the fourth row, I know there are 18. And then, of course, I really want to figure out the 20th row. All right. Well, this is an arithmetic sequence. The reason it's an arithmetic sequence is because there's a set number of seats more. In each row, or than the last row. So I add something to 6 to get here. I'm going to call it D for that common difference. Then I add the same number here, and I add the same number here. Now the neat thing about visualizing this is that now allows us to set up an equation. And what we can do is we can say that, well, when I take 6, and I add to it three of those common differences, I'm going to get 18. I can pretty easily solve this by subtracting 6 from both sides. Getting three D is equal to 12. 3D. Divide both sides by three, and I'll get D equals four. I want you to understand what we just found. We just found the common difference. We just found out what we had to add each time. What's cool about that is I can now kind of fill in the blanks. Which is a great check, right? 6 plus four is ten. Plus four is 14 plus four is 18. So it's a good check. Yet still, what I want is I want the 20th. Now remember for the 20th what we're going to do is we're going to start with that first one. And we're going to add 19 of those common differences. All right? That's going to give us 16 plus. Let's see, that would have been 80 76. And that gives us 82 seats in the 20th row. How cool is that? It's nice, huh? I hope you got 82. You may not have done it the way that I did it. But you probably have the same ideas in there. So it's a good problem to play around with. I'm going to clear out the text so pause the video now if you need to. Okay, here it goes. All right, let's finish up the lesson. So sequences are just lists of numbers. In a particular order, but their lists of numbers in a particular order. And arithmetic sequence is one where that list of numbers is always generated by adding the same thing to get the next number. Start with two, and I add 5. Then I add 5, then I add 5, then I add 5, et cetera, where I subtract three, subtract three, subtract three. Those are arithmetic sequences. All right? As I said later on in the course, we're going to see one more type of sequence called a geometric sequence. And in other courses, you'll see many other types. All right? But for now, let me just remind you this has been another E map instruction common core algebra one video. Until next time, my name is Kurt weiler. Keep solving problems, keep thinking, keep solving problems.