Formula for a Sequence
Date: 08/21/98 at 12:52:42 From: Jose Ignacio Alvear Subject: How to get a formula from a pattern Hey Dr. Math, I just received a math assignment which I think is too hard. The problem is as follows: term 1 2 3 4 5 n value 2 4 8 16 32 ? What I have to do is show the formula that is used to get the value, and I'm really having a hard time doing that. Jose
Date: 08/21/98 at 14:37:30 From: Doctor Jaffee Subject: Re: How to get a formula from a pattern Hello Jose, I'll try my best to help you out. There are a variety of techniques that you can use to solve problems like this one. Generally, the first thing that most people do is subtract adjacent numbers in the value row. If you keep on getting the same answer (which is not the case in your problem) then there is an easy way to arrive at the answer. For example: term 1 2 3 4 5 ... n value 3 7 11 15 19 In this case, the difference of adjacent numbers is always 4. That must mean that the value of the n term is 4*n + something. Since the value of the first term is 4*1 - 1 that must mean that the value of the n term is 4n - 1. You can check and see 4*2 - 1 = 7, 4*3 - 1 = 11, and so on. Now, if the difference between adjacent terms isn't the same in every case, but when you take the difference of the successive answers you always get the same answer, there is a slightly different technique. You would end up with a quadratic expression. But that's not the case for your example, either. But, the next method will work. Factor each of the values to prime factors. 2 = 2, 4 = 2*2, 8 = 2*2*2, 16 = 2*2*2*2, etc. In other words, your original problem could be rewritten: term 1 2 3 4 5 n value 2^1 2^2 2^3 2^4 2^5 ? (the ^ means exponent) Is the answer more obvious now? If not, write back and I'll try to help you some more. Also, send in any more problems that are giving you difficulty and I or one of the other Doctors will try to help you out. I hope my explanation has clarified the situation a little better. - Doctor Jaffee, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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