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Formula for a Sequence

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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

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/
```
Associated Topics:
High School Sequences, Series

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