Making a Series Sum to Zero

Date: 05/24/2002 at 13:01:01
From: Linda
Subject: 1,2,3 ........2005

How can I add only + and - signs between 1^2, 2^2, 3^2..... and 
2005^2 and make it finally equal zero??

Date: 05/24/2002 at 14:01:45
From: Doctor Douglas
Subject: Re: 1,2,3 ........2005

Hi, Linda,

This is a very interesting question!

First of all, there are 2005 numbers, 1002 of which are even and
1003 of which are odd.  The 1003 odd numbers must sum to an odd
number, and the 1002 even numbers must sum to an even number, so
the final total must be ODD, and cannot be zero.  But we can make
the final total equal to 1.

Let's use the following algebraic identity:

   (n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 

                 = 2n + 1

In other words, if you subtract adjacent terms, you will get this
progression (let's make sure that 2005^2 is the top of one of the
subtractions--this leaves the 1^2 at the bottom by itself):

  1    4     9    16     25    36     49  ....   2004^2     2005^2
 
  1     (9-4)      (25-16)      (49-36)   ....   (2005^2 - 2004^2) 

  1       5           9            13     ....        4009

You'll need to put a "-" in front of 4,16,36, etc., and a "+" in 
front of the other numbers to get this sequence (at least so far).

If we end up subtracting the 5 (i.e., 9-4), then we'll have to have 
a + in front of the original 4 and a - in front of the original 9.
(Does that make sense?)

Now see if you can figure out what to do with this last sequence
of numbers. Note that their differences are all four:  

  4 = 5 - 1 = 13 - 9 = ... 

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

Date: 05/24/2002 at 15:44:12
From: Linda
Subject: 1,2,3 ........2005

Dr.Douglas,

Thank you so much for your answer!!!