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Gauss and the Sum of Numbers from 1 to 100

Date: 03/02/2006 at 14:00:25
From: Danielle
Subject: Gauss's Trick

I was wondering how Gauss's trick works.  I know that he uses a 
variable to add up large sums of numbers like 1 - 100.

I am trying to understand this trick as a future educator, and learn 
how to break it down so I can teach the concept to my students.  I do
not really know where to start and would appreciate any help.

Thank you for your time.



Date: 03/02/2006 at 15:32:32
From: Doctor Peterson
Subject: Re: Gauss's Trick

Hi, Danielle.

I don't think we know just how Gauss himself thought through this 
problem as a child; we just know that he turned in his answer quickly 
enough to attract the attention of his teacher.  (I've seen many 
variations of the story, probably embellished with details we can't be 
sure of.)  It's worth noting, also, that the trick was not new to him, 
and was not a notable piece of math even at the time; it was just 
pretty good for a kid to come up with himself!

There are several ways to do it; you can use algebra, or pictures, 
or averages, among other approaches.  You can apply the formula for 
triangular numbers (which are sums of the form 1+2+3+...+n), which 
is a special case of the formula for summing an arithmetic series.

Here's one method I like:

Imagine adding up all the numbers:

    1 +   2 +   3 + ... +  98 +  99 + 100

Now imagine writing the same sum in reverse:

  100 +  99 +  98 + ... +   3 +   2 +   1

We can add up all 200 of these numbers (which will give us twice our 
sum) by pairing them off:

    1 +   2 +   3 + ... +  98 +  99 + 100
  100 +  99 +  98 + ... +   3 +   2 +   1
  ---------------------------------------
  101 + 101 + 101 + ... + 101 + 101 + 101

But that's just the sum of 100 of the same number, and is equal to 
100 times 101.  Our sum is half that, 50 times 101, and you can do 
that in your head: 5050.

Here's the picture version of the same idea:

  o o o o o
  o o o o *
  o o o * * 
  o o * * *
  o * * * *
  * * * * *

Here I'm combining two copies of 1+2+3+4+5, so I can find the sum by 
taking half of a 5 by 6 rectangle.  You can see here, by the way, why 
this is called a triangular number.

Almost equivalently, you can just pair up numbers at opposite ends 
of the list:

  1 + 100
  2 +  99
  3 +  98
  ...
  49 + 52
  50 + 51

There are 50 pairs, each summing to 101, so the total is 5050.  (This 
may well be what Gauss did, but I find it slightly more awkward than 
the first method, especially if there were an odd number to add up.)

Using the idea of averages, you can just note that the average of 
all the numbers is obviously halfway between 1 and 100, namely 
101/2; and the sum of 100 numbers with this average is equal to 100 
times their average, or 100*101/2, which again is 50*101.

I doubt that Gauss (or your students) would have used algebra, so I 
won't go into the algebraic versions.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Sequences, Series

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