Sum of an Arithmetic Series

(Special Case)

by Michael Steuben in Twenty Years Before the Blackboard

In elementary school, Carl Gauss (1777 – 1855) was asked to sum the numbers from 1 to 100. The teacher was probably expecting a few minutes of quiet, but Gauss produced the answer in seconds.

He probably divided the set of 100 numbers into 50 pairs of 101:

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101, etc.

Hence, the sum of 1 + 2 + 3 + 4 + … + 99 + 100 = 50 (101) = 5050.

Algebra students can now derive the formula for the sum of the first n integers

by using Gauss’ trick:

Let S = 1 + 2 + 3 + 4 + … + (n-2) + (n-1) + n

Then S = n + (n-1) + (n-2) + (n-3) + … + 3 + 2 + 1

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2S = (n+1) + (n+1) + (n+1) + (n+1) + … + (n+1) + (n+1) + (n+1)

2S = n (n + 1)

S = n (n + 1) / 2

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Now, back to Gauss, and the rest of the story …

School master: Class, I want each of you to spend the next 15 minutes summing the

integers from 1 to 100. This is difficult and I’m not sure of what

your chances are.

Gauss: Fifty-fifty?

School master: Incredible! You calculated the correct sum immediately!

You’re a genius!