Predicting the Next Number

Date: 8/30/96 at 10:44:46
From: Anonymous
Subject: Predicting the next number in a sequence

When given a series of numbers and asked to predict the next number, 
what is the formula for doing so?  Example:  2,5,12,23, ?

This question appears on psychological exams, federal employment exams 
and many others.  Is there a mathematical way of determining the next 
number in this series?

Date: 8/30/96 at 12:46:8
From: Doctor Jerry
Subject: Re: Predicting the next number in a sequence

First, if the first several terms of a sequence are given, there is no 
method for determining the general term. Suppose I'm given the numbers 
a, b, c, d and asked to determine the fifth and sixth terms of the 
sequence.  I'll show that any number of different solutions is 
possible.

I start by determining a polynomial 

   p(x) = Ax^3 + Bx^2 + Cx + D 

such that p(1) = a, p(2) = b, p(3) = c, and p(4) = d.  Then consider

   f(n) = p(n) + (n-1)(n-2)(n-3)(n-4) or
   g(n) = p(n) + (n-1)(n-2)(n-3)(n-4)(n-5).

Notice that both f and g determine sequences whose first four terms 
are a, b, c, and d.  Remaining terms are wildly different.  This idea 
could be elaborated.

You can, however, try to guess what was most likely in the mind of the 
person who made up the question.  For the sequences
2,4,6,8,...
1,4,9,16,...
I suppose most persons would say 10,12  or 25,36.

For the sequence you gave, 2, 5, 12, 23, I noticed that 5 - 2 = 3, 
12 - 5 = 7, and 23 - 12 = 11.  Since 3, 7, and 11 can be viewed as the 
odd numbers, leaving every other one out, one could argue that the 
next two terms are 23 + 15 and 38 + 19.  Other, more or less natural 
answers are possible.  However, one has no choice but to accept 
whatever the text makers decree is the correct answer!

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/