Gauss and Series Summation
Name: Joe N.
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.
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,
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.
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
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012