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Sum of Numbers from 1 to n

Date: 01/09/2003 at 03:43:03
From: Kobina
Subject: Summation of numbers

Hi Dr. Math,

Do you know if there is a formula for calculating the summation of 
numbers from 1 through n?

For example, let's say I want to add the numbers from 1 through 290.

Regards,
Kobina


Date: 01/09/2003 at 06:32:21
From: Doctor Jeremiah
Subject: Re: Summation of numbers

Hi Kobina,

Here is an easy way to remember how to do this:

Say S is the sum of the numbers from 1 to n. That means:

  S  =  1  +  2  +  3  +  ...  + n-2 + n-1 +  n   (n items)

But S could alos be the sum of the numbers in reverse order:

  S  =  n  + n-1 + n-2 +  ...  +  3  +  2  +  1   (n items)

But what would happen if we added these two things together:

  S  =  1  +  2  +  3  +  ...  + n-2 + n-1 +  n
  S  =  n  + n-1 + n-2 +  ...  +  3  +  2  +  1
 ===============================================
 2S  = n+1 + n+1 + n+1 +  ...  + n+1 + n+1 + n+1  (n items)

Now think about how multiplication is defined.

  2  +  2  +  2 is the same as 3 x 2
  4  +  4  +  4  +  4  +  4  +  4 is the same as 6 x 4
 n+1 + n+1 + n+1 is the same as 3(n+1)

And n+1 added to itself n times is n(n+1)

So that means:

 2S  = n+1 + n+1 + n+1 +  ...  + n+1 + n+1 + n+1  (n items)

is the same as:

 2S  = n(n+1)

Now we solve for S (which is the sum we want) and we get:

  S  = n(n+1)/2

Let me know if you have more questions about this.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Number Theory

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