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/