1, 7, 23, 55, 109, 191, ___

Date: 10/03/2002 at 10:54:23
From: Luke Wahlstrom
Subject: Number pattern

My family is stumped on this number pattern. We have tried every way 
we know to solve it. Could you help us out?

1,7,23,55,109,191,___  (fill in the blank)

Thanks!
Luke

Date: 10/03/2002 at 15:48:15
From: Doctor Greenie
Subject: Re: Number pattern

Hi, Luke -

Often a sequence like this is created from a formula in the form of a 
polynomial. If that is the case, the formula can be found by applying 
a process called the method of finite differences. Here is a link to a 
page in the Dr. Math archives where there is a detailed discussion of 
this method:

   Method of Finite Differences
   http://mathforum.org/library/drmath/view/53223.html 

This method is rather advanced for most 11-year-olds; you (or you and 
your family) are certainly welcome to give it a try if you want to.

By just playing around with some ideas, I discovered a way you might 
be able to find the formula for generating the terms of your sequence 
without using a method as complicated as the method of finite 
differences. To be able to do this, you will need the following hint:

Compare the given sequence of numbers to the cubes of the first 
several integers, as indicated in this table:

    integer n  n cubed   term in sequence   difference
   ----------------------------------------------------
        1          1               1             0
        2          8               7             1
        3         27              23             4
        4         64              55             9
        5        125             109            16
        6        216             191            25
       ...

The differences between n cubed and the n-th term in your sequence 
show a nice pattern which you can use to develop the formula (or rule) 
for producing the terms of your sequence.

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/ 

Date: 10/04/2002 at 11:42:08
From: Luke Wahlstrom
Subject: Thank you (Number pattern)

Dr Greenie:

WE GOT IT! Thank you so much. It was really hard for me and my family.  
We got an answer of 307. (1,7,23,55,109,191,**307**). You were very 
helpful. I'm glad we didn't have to use the "advanced" method.

Luke