Finding Formulas for Number Sequences
Date: 11/22/97 at 02:10:07 From: Ricky Chu Subject: Formula Hi Gentleman, My question is about trying to find a formula between numbers. For example, suppose I have 10 numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 It is obvious that the first number plus 2 will yield the second number, 4; 4 plus 2 will yield the number 6, and so on. So the formula for this group of numbers is x + 2. Say I have another set of numbers: 2, 6, 14, 30, 62..... If the first number is multiplied by 2 and 2 is then added, you will get 6. 6*2 + 2 will yield 14. 14*2 + 2 will be 30.. So the formula is x*2 + 2. The above two examples are very simple. The formula can be found by observation. However if I have a group of numbers like 13, 80, 145, 167, 188... Those numbers look random but they are related. Do you have any idea how to find the formula? I only want to know the method. I have tried to use the difference equation, but it doesn't work. Best Regards, Ricky.
Date: 11/22/97 at 05:40:52 From: Doctor Mitteldorf Subject: Re: Formula Dear Ricky, Finding numbers from a formula is a science. You can even program a computer to do it. But finding a formula from numbers is an art. You try this and you try that, and with a little luck you're sometimes successful. You've mentioned differences: taking successive differences is a powerful way to give you insight into the formulas behind many sequences. Taking ratios of successive terms, dividing each by the previous one, is another technique. Remember that there's no one "right answer" for finding a formula to match a sequence of numbers. That's because there are a lot more formulas than there are sequences, and you can always find several formulas that work. In test questions, there's usually one that "seems" simplest, and that's what they're looking for; but it's not the only answer... Here's one "brute force" way that will always generate a formula, but it might not be the "simplest" formula. If you have 5 numbers in the sequence, they can be matched by an expression of the form: a + bn + cn^2 + dn^3 + en^4 Just write down the 5 equations and solve for a,b,c,d and e. For the sequence you suggested, the n=0 term is 13, so a=13. The next term is n=1, so a + b + c + d + e = 80 The next term is n=2, and the corresponding equation is a + 2b + 4c + 8d + 16e = 145 Try writing down the other two equations and solving them all together. You'll get a formula for sure, but it may not be the simplest or most satisfying one. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 11/24/97 at 01:34:26 From: RICKY CHU Subject: Further problem on formula Dear Doctor Mitteldorf, Thanks for your help and ideas. I have tried your suggestion to find out a formula, but this is not the one that I want. I used a group of numbers, say 10, 18, 34, 66, 130. The formula is (x * 4 + 2) / 2 - 3. That formula can be simplified as (x * 2 - 2). If I use your suggested solution: a + b*n + c*n^2 + d*n^3 + e*n^4 = 10, a = 10; b = 14/3; c = 11/3 d = -2/3; e = 1/3; I agree with you that there are many formulas to generate the same result. My further problem is I have to use the existing data to forecast/predict the following numbers. If I use the above approach, I can't find the following numbers. How do I know the number next to 130? Do you have any idea to find the mentioned formula or any book that you can refer me to? Best Regards, Ricky.
Date: 12/03/97 at 04:51:21 From: Doctor Mitteldorf Subject: Re: Further problem on formula Dear Ricky, The solution you found is certainly simpler and more elegant than mine. You can always match a list of 5 numbers with a formula that has 5 free parameters in it (a b c d e). Your formula has only 2 free parameters and it works - but this is a measure of 1) the fact that your formula is "better," and 2) that there really is a pattern in your numbers. A geometric sequence is one in which each number is a constant times the previous number. They are not hard to detect. But your sequence is "almost geometric." I suppose the best way to detect it is to notice that each number is close to twice the last, and that might inspire you to play with the disparity. In general, pattern recognition is an art as well as a science. With experience, you just get better at it. Does the sequence in your question come from "real" data, or is it a made-up problem? For made-up problems, it's anyone's guess; but if the sequence comes from some scientific or mathematical source, the source must be considered a valuable clue to what kind of pattern to expect. You can search for resources on this topic under "pattern recognition" and under "sequences". Both terms are subject to lots of distraction: "pattern recognition" is also used to describe the process of recognizing images, like people's faces, etc, from a collection of dots. "Sequences" can also describe the area of mathematics that relates to summation of infinite lists of numbers. Here are two resources I turned up with a search: Andrews, Harry C. Introduction to Mathematical Techniques in Pattern Recognition. Malabar, FL: Krieger, 1983. 242 pp. Sloan's On-Line Encyclopedia of Integer Sequences http://www.research.att.com/~njas/sequences/index.html -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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