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Sequence of Squares


Date: 07/25/98 at 23:52:11
From: Sam
Subject: Geometric sequences

I have searched the Internet but cannot find any information 
specifically discussing or defining the sequence of squares. I would 
be very grateful for this. 

Thanks, Sam.


Date: 08/04/98 at 16:18:18
From: Doctor Margaret
Subject: Re: Geometric sequences

Hi Sam, 

Thanks for writing to us. A sequence of numbers is a set of numbers 
arranged in an orderly fashion such that the preceding and following 
numbers are completely specified. For example, in the sequence:

   ..., 7, 10, 13, 16, 19, ...

the difference between the numbers is 3, and this shows us that the 
number preceding 7 is a 4 and the number after 19 is 22. Take a look 
at these numbers:

   1                          = 1
   1 + 3                      = 4
   1 + 3 + 5                  = 9
   1 + 3 + 5 + 7              = 16
   1 + 3 + 5 + 7 + 9          = 25

So far, the righthand column turns out to be the sequence of perfect 
squares. I think it's neat that this sequence of squares is made up of 
consecutive odd integers (the lefthand column ). There is also a way 
to express this algebraically.

Odd integers are usually written as (2n - 1) or (2n + 1). The 2 makes 
sure that the first number is even. The n stands for the number of 
terms in the sequence, for instance, for three terms starting at 
n = 1:

    3
   ---
   \    2n-1 = 3^2 = (2(1)-1) + (2(2)-1) + (2(3)-1) = 1 + 3 + 5 = 9
   /
   ---
  n = 1

You don't have to start at 1. You can start anywhere and go on for 
however many terms you like or are required to. 

Let me know if you need anything else on this subject. Thanks for 
writing.

- Doctor Margaret, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Sequences, Series

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