Finding a Rule for a Sequence

Date: 07/24/2003 at 13:50:59
From: John
Subject: Lots of trouble on this one

I have to find the next number and it's a mystery to me. 

1, 3, 11, 67, ?

Date: 07/25/2003 at 01:34:24
From: Doctor Greenie
Subject: Re: Lots of trouble on this one

Hi, John -

It is impossible to determine what THE next number is in this 
sequence. Given ANY sequence of numbers, it is possible to develop 
many different rules which produce that given sequence of numbers - 
but those many different rules may well produce different "subsequent" 
numbers. In problems like this, as the number of given numbers 
increases, it becomes harder and harder to find a rule that fits all 
the given numbers; but with only the first four numbers given, it is 
comparatively easy to find a rule that works.

Here is one way to find a rule that will find the next number (and 
subsequent numbers) in the sequence:

   1  ==>  times 2,  plus 1  ==>  = 3
   3  ==>  times 4, minus 1  ==>  = 11
  11  ==>  times 6,  plus 1  ==>  = 67

And so the next number in the sequence will be

  67  ==>  times 8, minus 1  ==>  = 535

However, as I have noted, this is only one possible answer to the 
problem. It is quite possible to find completely different rules 
that generate these first four given numbers but give very different 
next numbers.

I hope this helps. Please write back if you have any further questions 
about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/