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Sum of Numbers 1-500


Date: 06/20/2001 at 18:02:44
From: Mary Griffing
Subject: Addition

My question is, what is the formula to find the sum of the numbers one 
to five hundred?


Date: 06/20/2001 at 18:34:45
From: Doctor Jodi
Subject: Re: Addition

Hi Mary,

Have you heard of a mathematician named Gauss? When he was about 10, 
his schoolmaster gave the following arithmetic problem: sum the 
numbers from 1 to 100. Almost immediately, Gauss placed his slate on 
the teacher's desk, indicating that he had finished the problem.

His teacher, of course, was incredulous. When he discovered, though, 
that Gauss' answer was correct, the teacher knew that he could not 
teach Gauss mathematics and arranged for the boy to receive math 
lessons from his assistant, a youth of about 17.

How did Gauss solve this problem?

Let's first add the numbers from 1 to 10. Let's write the numbers this 
way:

 1    2    3    4    5
10    9    8    7    6  

Do you notice anything useful about this arrangement?


 1    2    3    4    5
10    9    8    7    6  
--   --   --   --   --
11   11   11   11   11

Each column sums to 11, and there are five columns so the sum we want 
is 55. Notice that 11 is 1 more than 10, and that 5 is half of ten. So 
we took (10+1) * (10/2)

Let's try the same trick with the numbers from 1 to 100 (we'll 
abbreviate of course):

  1  2  3  4  5  6  7  ....  45  46  47  48  49  50
100 99 98 97 96 95 94  ....  56  55  54  53  52  51

Each column sums to 101 and there are 50 columns, so the sum we want 
is 5050. That is, (100 + 1) * (100/2).

Does this make sense?

Let's summarize our trick. If you want to add up the numbers from 
1 to 2n (where 2n stands for any even number), you just multiply 
(2n + 1) * 2n/2, i.e. (2n + 1) * n.

Now can you solve your problem of adding up the first 500 numbers?

It's similar, but not _exactly_ the same if you want to add up the 
first n numbers where n is odd. For the first 9 numbers, you would 
look at

1  2  3  4  5 
9  8  7  6

to find a pattern. What pattern do you find?)

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