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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

- 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

Luke
```
Associated Topics:
High School Puzzles
High School Sequences, Series
Middle School Puzzles

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