Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Gauss and Series Summation
Name: Joe N.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 

What is the total of the numbers 1 through 100 (inclusive). I understand that Gauss had the formula for doing this calculation very quickly. What is the formula and does it have a name.

Hi: The story is presented as follows: Gauss as a young school kid was being punished by his teacher for being disruptive in class. The punishment was to add the numbers from 1 to 100. Gauss blurted out the answer. The formula, if you will, is to add 1 +100, 2+99, 3+98, ...48+53, 49+52, 50+51.
So, we have the number 101 fifty times or 5050.

Dr. Harold Myron

Yes Gauss did! The story (I do not know whether it is true or not.) is that
when Gauss was in the third or fourth grade (I do not know exactly, but a
very early grade.) his teacher gave this boring assignment to his class. (I
think that "busy work" is not a new phenomenon.). Within a minute, to the
amazement of the teacher, Gauss turned in his paper with a single number
written on it (2525). Here's how Gauss thought:
     1   +    2   +     3      +    4    +....  +   98   +   99   +   100
100  +   99    +  98      +  97    + .... +     3   +     2   +       1
101     101       101      101       101      101       101      101

This sum is:    100 x (101) = 10100. But this is adding the sum from 1 to
100 two times, once going up and the second time going down, so 10100 must
be divided by 2. So the number Gauss wrote was:
     10100 / 2 = 5050 !!
The book, "Mathematics -- from the Birth of Numbers" by Jan Gullberg has a
nice discussion of the topic of  the sum of finite and infinite series.

Vince Calder


The proper name for this is a "series". A series is a sequence of numbers added together. The formula is based on the fact that you can express the series as sets of 101. The first number plus the last number is 1+100=101. The second number plus the second-to-last number is 2+99=101. The third plus the third-to-last is 3+98=101. This continues on until 50+51=101. There are fifty sets of (101). The formula for the sum of all numbers from 1 to N is sum=N(N+1)/2. I don't know whether it has a fancy name. The sum is 5050.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory