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College Algebra: Understanding Sequence Problems

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:23
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 90 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Further Topics (12 lessons, $17.82)
College Algebra: Solving Sequence Problems (3 lessons, $4.95)
College Algebra: Sequences and Series Intro (3 lessons, $4.95)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

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 UT-Austin 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, p-adic 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.

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Now a lot of times we're looking for patterns. In fact, really mathematics, if you think about it, is just the study of patterns and seeing how things are changing and how things are moving and then what patterns can arise from those things. Well, in fact, at best the good way to first see these patterns in action is to actually look at sequences of numbers. So it's just a whole bunch of numbers that are given to us and then we have to see if we can find some pattern that leads through and makes these things apparent what the pattern is. And so such objects are called sequences. These are just collections of numbers that appear in a particular order so an ordered collection of numbers go off.
The simplest and most familiar sequence is just the numbers that we count with, 1, 2, 3, 4, 5, 6 and so forth. But, of course, there are a lot of other sequences. In fact, there are tons of sequences. There are, for example, sequences that might look like this - 1, 2, 4, 8, 16, 32 and so on. Those aren't just all the numbers put together but, in fact, you can see the pattern, it's just all the powers of 2. Or I take the previous answer and multiply it by 2 and get the next one. Take this and multiply by 2 and get 4. Take this and multiply by 2 and get 8 and so on. There's a pattern there to that sequence. Some sequences, by the way, have no pattern at all. I mean you can just make up a random sequence, 1, -3, 9, 7, 103, 2, and there's no discernable pattern to that at all. So not every sequence has to have a pattern. But sequence is just an ordered list of numbers, a collection of numbers in order.
Now sometimes you actually want to be able to express a sequence if it does have a nice pattern in some way. And we can actually use a formula like we use for functions, we can use that actually for sequences. Now I want to show you how this would go. So, for example, here's an example of a sequence I'm going to describe to you just using a formula. So I just want to show you the notation here really. So I would say that the sequence would be called An. That's the nth term in the sequence. So this sequence would actually look like, well in the first term A1 and then A2 and then A3 and then A4, and so on. So the little subscript just tells me which term of the sequence I'm in. So this is the 4th term because there's a little 4 there, that's all that means. So this is the first term of the sequence, this is the second term of the sequence, and third term and so forth. Now what's the formula for this? Well as an example, and this is just an example, we could say something like A1 = 1. So I tell you the first term is one in this case. And then to find out all the other terms you can generate them by letting me know what the nth term is, the general term. Suppose that's 2n - 1. Now let's take a look and see if we can now generate the other terms. Well, for example, how would I generate A2? Well, it's just like a function. Wherever I see an n I put in 2. And so what I would see is, we'll put in 2 here, I would see that A2 would equal 2 x 2 - 1, which would be 4 - 1, which is 3. What would A3 be? Well, again, think of function. Wherever I see an n, I plug in 3. So I put a 3 in here and I see 2 x 3, which is 6 - 1 is 5. If I put in a 4 in here for n the fourth term would be 8 - 1 which is 7 and so forth. So, in fact, this is a very compact way of actually giving a formula for an entire sequence and it's very much like a function, right? A is the function of n, where now n is the placeholder, it tells me which particular point in the sequence I'm at. For example, if I wanted the 20th term in this sequence I would put in a 20 here and I'd see 2 x 20 - 1 and so I'd see 40 - 1, which would be 39. So this is just a compact way of writing out a sequence that actually behaves in a certain way.
Maybe I can give you one more example. Sometimes sequences require the previous numbers of the sequence to figure out the next number. So for example, suppose I tell you that another sequence has its first term 1 and then to get its nth term you'd take 2 times the previous term and you add 1. What does that sequence look like? Well this is a little peculiar and we'll be careful here. So let's list those terms down in order. So what would A1 be? Now remember, the sequence I just write as A1, then A2, then A3, then A4. I just write them out in order where the little subscript tells me which term in the sequence I am. So A1 is 1, we were told that. Now how do I find A2? Well, A2 would equal 2 x A, and what subscript is this? 2 - 1 is 1, so it's just the previous number, 2 times the previous number plus 1. So 2 times the previous number would be 2 + 1 would be 3. So, in fact, what I'd see here is that A2 = 2 x A1 + 1. And where'd that 1 come from? That's just 2 - 1. Wherever I see a 2, I'd replace the n by 2. So 2 - 1 is 1. So I need the first term, which is 1. So 2 x 1 + 1. What would the next term be? Well, A3, if I let n be 3, I would see 2 x A3 - 1, which is 2 + 1. So I need the previous term, which is 3, multiply it by 2, I get 6 + 1 = 7. You see, I'm actually able to generate the sequence, but I always have to know what the previous number was. What would this be? Well, this would be 2 times the previous thing, so that would be 14 + 1 = 15. What would this be? This would be 2 times the previous thing, which would be then, 30 + 1 = 31. This is sometimes called an Inductive Definition because you can generate these things inductively one after the other, after the other, after the other and so forth.
Let me show you a very famous such sequence. Let's let A1 = 1. I'm even going to tell you what the second one is. It equals 1 as well. But to get all the rest, what you just do is add the two previous sequels. So let's make a list of the terms. I have 1, 1, and what's the next one? Well, for the next one I just add the two previous sequels, so I add 1 and 1 and I get 2. To get the fourth term, what do I do? I add the two previous sequels, so I add 1 and 2 and get 3. The next term I'd see 5. The next term I'd see 8. The next term I'd see 13, and this goes on. This sequence has a name; this is called the Fibonacci sequence. And, in fact, it's a really, really cool sequence because, in fact, if you ever take a look at the spirals on a pinecone or on a flower or on a pineapple or anywhere, if you count the spirals, and there are two sets of them, there are some that are going this way and there are some that's going this way. You count them; you always see two adjacent numbers. Like for a pineapple, if you have a pineapple you'll see 8 and 5, 8 spirals going in one direction and 13 spirals going in the other direction. That's right, 8 and 13. If you look at a pinecone you'll see 5 and 8. So it's really cool. In fact, sometimes these sequences actually capture some essence of nature and are reflected back into our real world. So these things are really fun and important. The important thing right now is to realize this is not a very big deal, it's just a little notation that may look scary, but it just allows us to find the nth term in a sequence.
Up next we'll take a look at some very special sequences and some properties about them. I'll see you there.
Further Topics in Algebra
Sequences
Understanding Sequence Problems Page [2 of 2]

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