The Punishment of Gauss and Arithmetic Series

Friedrich Gauss was a German mathematician who derived the equation for the Bell Curve, which is also called the Gaussian Distribution. His mathematical ability was first recognized when he was in elementary school. 

Gauss had been falling asleep in class, so the teacher decided to punish him by telling him to find the sum of all the numbers from 1 to 100. The teacher was stunned when Gauss gave the correct answer after only thinking for a few seconds. Take a few minutes to see if you can figure it out on your own.

In the set of numbers from 1 to 100, there are pairs that add up to 100: 1 and 99, 2 and 98, etc. There are 49 such pairs. 50 is in the middle and 100 is at the end. Put that all together, and we have 49 pairs that add to 100 plus 100 to get 5000 and 50 more gives the correct answer of 5050.

A series like 1, 2, 3 is called an arithmetic series because the difference between any two consecutive terms is the same. Suppose we want a general formula for the sum of any arithmetic series. Let S be the sum of such a series. If a_0 is the first term, d is the common difference, and n is the number of terms, the equation for the sum of the series is:

S = a_0 + d + a_0 + 2d .... a_0 + nd

We can also write this equation in reverse order starting with the nth term.

S = a_n + a_n - d + a_n - 2d .... a_n - nd

If we add these two equations to together, the positive d terms in the first equation cancel out the negative d terms in the second equation. We also have a_0 n added to itself n times and likewise with a_n That leaves us with:

2S = n*a_0 + n*a_n

Simplifying, we get:

S = (n/2)(a_0 + a_n)

To put it another way, the sum of any arithmetic series is just the average of the first and last term multiplied by the number of terms. Let's check that with problem Gauss did.

(100/2)(1 + 100) = (50)(101)

(50)(101) = 5050

We get the same answer as Gauss. The formula for the sum of an arithmetic series can also be written as

S = (n/2)(a_0 +a_0 + (n-1)*d)

Let's double check that.

S = (100/2)(1 + 1 + (100 - 1)*1)

S = (50)(2 + 99*1)

S = (50)(101) = 5050