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About this Lesson
 Type: Video Tutorial
 Length: 10:00
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 107 MB
 Posted: 06/26/2009
This lesson is part of the following series:
College Algebra: Sequences and Series Intro (3 lessons, $4.95)
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/intermediatealgebra. The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
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We've been thinking about functions for a while  you know, the idea of you have f of x equals something, and you can plug something in for x, and then something else will come out for y. For example, a real quickie example would be something like f(x) = 2^x. And if you plug a number in here  I just plug it in for x, and I'd see what it equals, and we're on our way. Great. Now there are times when in fact I want to restrict the numbers that I'm plugging in. I want to restrict sometimes what's called the domain to just the counting numbers  1, 2, 3, 4, 5, and so forth. For example, if I did that, I could say f(1) and that would be 2^1. I could say f(2), and that would be to the 2^2. I could say f (3) in this example, and I'd see to 2^3. And you can compute these things: 2, 4, and 8, and so forth.
Now, when I have functions that I just want to restrict the values I plug in to be the counting numbers, we actually call such things sequences. And we denote them in sort of a funny way, and let me show you the funny way that we denote them. So these are really sequences; and the sequence is just a function where I'm just going to plug in natural numbers. Instead of writing it like this, I can write it like this: a[n]  this is new notation  = 2^n. And that is actually saying the same thing as this, but the notation's all new, so I want to go through this with you really slowly.
See here, this is the variable, and the f tells me  this is like the map, and the map is the 2 to the power x, and x is the variable. Here, to translate that, a is sort of like the f here; a is telling me that this is the 2 to a power thing, and this little index here, this n, represents the variable. So, for example, if I plug in a natural number in for n  "n" for natural number  then I would plug that value in here So the analog of this statement would be a[1]. That means let n = 1, just like here I'm saying let x = 1; and the answer would be 2^1. Wherever I see a 1, I replace that 1 in for the n. So I see 1 here  n  I put it here; a[2] would equal 2^2; a[3] would equal 2^3, and so forth.
So this is a way for me to describe a function where the function now is only going to take on  I'm only going to plug in values that are counting numbers  1, 2, 3, 4, and so forth. And this is how it looks, so these three things are the exact same thing as these three things here; but it's a different notation. And the reason why we like this notation is because just by indexing them  and now they come in an order  there's a^1, a^2, a^3, a^4, a^5, just like the counting numbers come in order; and I can see that order by listing them as a little subscript here. Whereas here it's hard to write all that stuff with the parentheses and everything; I can just restrict myself to this language.
So let's take a look at examples just looking the notation. So it's like learning a new language; once you get into it, you'll be speaking it really fluently. So, for example, let's take a look at this sequence: the sequence defined by a[n] = . Now that's the rule. You could think of that like a function. It would be like f(x) = .
Now let's see if we can write down the first few terms in this sequence. Since the natural numbers come in a particular order  1, 2, 3, and so forth  this sequence comes in a particular order  a[1] would equal what I get when I plug in 1 for n. So I'd have , which equals 1. So the first term in my sequence is 1. What's a[2]? A[2] I plug in 2 for n, so I see . And what does that equal? That equals zero. What about a[3]? Well, wherever I see an n, I'm now going to plug in 3. So I see , which equals . And you can keep doing this. For example, what is the 11th term? I can fast forward right to the 11th term, because I'm just saying n is 11. So this would be , which equals what? Well, 11  2 is what, that's like 9, over 11. So the 11th term in this sequence is actually . You see how I'm evaluating the terms by just plugging in the appropriate value for n into this thing. It's just like a function, just written differently. It allows me to go through and look at these things.
Now let's try another example. Let's try an example now where I'm going to give you the first few terms, and let's see if you could find a pattern. So here's one. How about I tell you that a[1] is 5, a[2] is 10, and a[3] is 15. Can we guess a pattern? Can we guess a general formula? Well, it seems like these are the just the multiples of 5. And so a good guess might be a general formula, a[n] would just be 5 times n. And let's check it. If n = 1, I get 5; if n = 2, I get 5 times 2, which is 10; if n = 3, then I have 5 times 3, which is 15. So if you see the first few terms of a sequence, we can actually try to produce a general formula that will give us the sequence for any n.
How about this one here: is the first term, so that's a[1]; a[2] = ; a[3] = ; a[4] = . Can we see a pattern here? Well, let's see. It seems that the tops are just being increased by 1, and the bottoms are also being increased by 1. So what would be a general formula  "a[n] =." Well, the top always seems to be the same as the index. I have a sub 1 here, and there's a 1; subscript 2, and I have a 2; subscript 3, and I have a 3. So it looks like if I have a subscript n, I should put an n here. And on the bottom, I just have one more than the top. That would be n plus 1. And that's how I'd write this, what really is a function, but I'm now thinking about it as a sequence, so I'm only going to let n be 1, 2, 3, 4, and so on, all the way down the line. So now we see a general formulation for the beginning part of this sequence.
Now there are other ways of writing sequences. And this is sort of an important idea. We can write sequences in a recursive manner. These are called recurrent sequences, or recursive sequences; and they're called recurrent sequences because, to find the next term, you have to actually use the previous term somehow. So let me show you this with an example. Suppose I say that the first term in our sequence is 1, and then here's the general rule for the nth 1: You take 2a[n1] + 3. Now that looks really complicated, so let's think about what this means. To get the next term, what I do is take the previous term  notice that's the term that would come right before this one. If this is a[n], the term that comes before it would be a[n1]. So this is the previous term, I multiply it by 2, and then add the number 3. Now what would these look like? Well, a[1] we know is 1, that's a given. Now what about a[2]? Well, now a[2] would be a[1] times 2 plus 3. So I plug in a 1 here, and I'd see 5. Now how would I find a[3]? Well, a[3] = 2a[2] plus 3. So now I use this answer, and I plug this answer into here. You see how I keep using the previous answer to help me find the next one? Here I plug this in, I see 2 times 5, which is 10, plus 3 is 13. A[4], what do I do now? Now I take the 13 and plug that into here, take the previous one, multiply it by 2, that would be 26, and I add 3, which is 29, and so on. So this is called the recurrence, or recursive sequence, because, in fact, I have to use the previous person and do something to it in order to get the next person, or to generate the next person.
And the most  let me show you now the most important, or the most famous recurrent sequence. The first term will be 1, the second term will be 1, and then from that point on, if you want to get the next term, you just take the previous term and add it to the term before that. So these are the two previous terms I'm adding together to get the next term. Let's make a list of what that looks like. First I get 1, and then I get 1 again, and now what do I get? Well, to get a[3], I take a[1] + a[2], the two previous terms. So I add these two up, and I get 2. To get the next term, I add the two previous terms up, and I get 3. I add the two previous terms up to get the next one, which is 5, then 8, then 13, and these are known as the Fibonacci numbers, probably the most famous recurrent sequence that's out there  you add the two previous terms together to generate the next term. Anyway, this is the idea of sequences, and the notation is a little bit new and different; it takes us a while to adjust to it. But once we think about it as a list of numbers in a particular order, we're going to be home free. I'll see you up at the next lecture.
Sequences and Series
Sequences and Series
General and Specific Terms Page [1 of 2]
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