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College Algebra: Problems with Geometric Sequences

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  • Type: Video Tutorial
  • Length: 10:52
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 116 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)

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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Okay, so Arithmetical Sequences are sequences where, in fact, to get the next term what you do is just add a constant number every single time and that keeps building up the sequence.
Let's take a look at another type of sequence that has the same spirit, but different arithmetic. Here's an example: Suppose I start with the 3, and then I see a 12, and then I see a 48, and then I see 192. Now you can see immediately this sequence grows a lot faster then any kind of Arithmetical Sequence and so can't be because here to get to here I'd have to add 9, but here to get to here I have to add a whole bunch of stuff. I have to add 36 to get from there and so on so that the amount I'm adding seems to be increasing. So it's not going to be an Arithmetical Sequence. Is there any pattern to this? Well, you may notice that, in fact, to get from here to here, I multiplied 3 x 4. Now how do I get from here to here? Well, notice if I take this and multiply it by 4, I actually get the next term. And if I take this and multiply it by 4, I get the next term. So, in fact, this does have some sort of pattern. It has a multiplicative type structure where to get the next term, instead of adding a constant number I'd multiply by a constant number. These are called Geometric Sequences. And, in fact, these are sequences that serve really in some sense explain growth. In fact, we've seen a lot of these things in some sense when we think about interest rates and compounding and things of that sort. The essence of it really is a geometrical type sequence where things grow very quickly by multiplicative factor.
So, how could I describe a Geometric Sequence in general, but what property must it have? Well, it must have an order for me to get the next term, I'd take the previous term and multiply by a constant number, a constant ratio. That means that if I take any two consecutive terms and divide this one into that one, the answer should always the same, it should be that ratio, and so what we say is a Geometric Sequence is one where we have a constant ratio. If you take the nth plus 1st term and divide it by the nth term, it's always the same. Look at it. If I take 12 and divide it by the previous term, 3, I get 4. If I take 48 and divide it by the previous term, I get 4. If I take 192 and divide it by the previous term, I get 4. So it's always the constant ratio, it's always the same. So this, in fact, explains and defines a Geometric Sequence.
Well, now the question is, how can you now give a formula for a Geometric Sequence that you want to find the nth term? Well, let's look at this example and see. So this is going to be 3, and let's write this out, this is going to be 3 x 4, and then to get this term, what do I do? I take the previous term and multiply it by 4. So now it would actually be 3 x 4^2 wouldn't it, because 4 x 4, and this would be 3 x 4^2 x 4, which is 4 cubed. And then I see 3 x 4^5, and so on. So let's see, if this is the 1^st term, and this is the 2^nd term, and this is the 3^rd term, and this is the 4^th term, and this is the 5^th term, what's the pattern? Well, the pattern is, let's see, if I want to get the nth term, what do I do? Well, I always seem to have the 1^st term as a multiplicative factor everywhere. See how the A^1 is in every single thing, the 3 is in every single thing, so it has an A^1, and then I'm going to have the r, that ratio, that common ratio to some power. Now what power should it be? Well, in the 2^nd when n = 2 the power should be 1. When n = 3, the power should 2. When n = 4, the power should be 3. When n = 5, the power should 4. So what's the pattern? Well, the pattern is if I'm at n, the power should be 1 less than n. In each case 4 and I'm at 3, 3, I want exponent of 2 and so on. So it should be n - 1. So there's actually a formula that will generate the nth term in a Geometric Sequence starting with a 1 and having that common ratio, r, so you can actually find any sequence you want.
For example: Suppose that I tell you that I'm thinking of a Geometric Sequence, its 1^st term equals 4 and its 2^nd term equals 20. Can you tell me what the 4^th term is going to be? Well, to find the 4^th term I've got to find that ratio because I already know A^1, but I need to find that ratio. How could I do that? Well, if I plug this in to this formula, what would that give me? Well, that would tell me the following: It would tell me that A^2, A^2 which I'm told was 20, so 20 would have to equal A^1, which I know is 4, times this mysterious ratio, which I have no idea what that is, but raised to what power? Well, now I raise it to the n, which is 2 - 1, 2 - 1 is just 1. So I can now actually solve this for r. If 20 = 4r, then I know that r must equal 5. And so then the formula in this case, for this particular example, would look like this, A[n] = 4, because that's the A^1, x r, which is 5, raised to the power n - 1. So there's the formula. So if you want to know A^4, that's real easy. I know the answer. The answer would now be, 4 x 5, raised to what power? Well, the 4 - 1, which is the 3^rd power. So that would equal 4 x 125, which equals what? Well, that equals 500. So the 4^th term in this geometric series is already 500. We start with the 4, go to a 20, and we're already at the 500 in the 4^th term, a really fast growing sequence.
Let's try one more example just to illustrate this way of thinking. Suppose I tell you I have a Geometric Sequence, the 3^rd term is 5 and the 8^th term is. And my question is, give a general formula that generates all the terms, so what's the formula for this Geometric Sequence? Well, the first thing you notice is that if A^3 = 5 and the A^8 is the small number, in fact, that ratio probably is going to be a fraction that's going to be less than 1. Because I have to multiply 5 by something that's less than 1 to make it small, and smaller, and smaller to get it down to here. So, in fact, this is going to be a sequence that's going to be decreasing rather than increasing. But any case, we know the formula. The formula is, the nth term of the sequence is just A^1 x r^n-1, so let's plug in this information. When n = 3 I know that A^3 = 5. So I have 5 = A^1, don't know what that is, r to what power? Well, if n = 3, 3 - 1 = 2. So there's an equation that's in A^1 and r, and I don't know either of them. That's the problem. All right, now one more fact. I know that when n = 8, this number is , and what does that equal? Well, it equals A^1 x r, to what power? Well, if n = 8, then this would be n - 1 = 7. But look what I have; I have two equations and two unknowns. I can find this again. Now it's not linear. It's not linear because I have exponents, so you can't use any matrix stuff now. But what I will do is, I'll use the substitution method. In fact, what I'll do is pick this and solve it for A^1 and substitute that value in here. I solve this for A^1, what I see is A^1 = 5 ÷ r^2, and if I now take that and plug that in for the A^1 here, because that equals A^1, then what do I see? I see =, now instead of A^1 I'm putting in 5 ÷ r^2. So I see 5 ÷ r^2, and I multiply it by r^7 so I can actually cancel, I have two rs down here, I have seven rs up here, so if I cancel, I'll be left with r^5, and so now if I divide both sides by this 5, I see that r^5 = what? Well, I divide both sides by the 5, I see x another 5 down there, which is . Well you can take 5^th roots of those sides and see that, in fact, r has to be 1/5^th. That is if you take 1/5^th and multiply it by itself 5 times you actually get . You can check that. So there's r, there's that constant ratio, what's A^1? Well, I could find A^1 by just plugging back into here, so A^1 would = 5 ÷ r^2. So that would be 1/5^2. Well, 1/5^2 is actually 20 is , but that's a complex fraction, so I invert and multiply so I get the 25 on top so I see 5 x 25, which is 125. So that's the 1^st term and every successive term I take the next term and I multiply it by 1/5^th, take that answer and multiply it by 1/5^th, and so on. So a general formula I can now report is A[n] = 125 x 1/5^n-1. And that will generate now every single term in this Geometric Sequence. If you want to know the 27^th term, you just plug in n = 27, and I see 125 x 1/5^26, 27 - 1.
So, again, just like with Arithmetical Sequences, Geometric Sequences you can always find out exactly the whole sequence just knowing two pieces of information. Either the 1^st term and that common ration, or just two terms in the sequence and you can then solve, again two equations and two unknowns, for r and the 1^st term.
Geometric Sequences, I love them. See you soon.
Further Topics in Algebra
Sequences
Solving Problems Involving Geometric Sequences Page [2 of 2]

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