RedTailHawk on Nostr: https://m.primal.net/PGRf.jpg One day as a schoolchild, a boy who would go on to ...

One day as a schoolchild, a boy who would go on to renown as one of the greatest mathematicians of all time, Carl Friedrich Gauss, blurred his way through some scholastic busywork assigned by his schoolmaster.

My personal speculation is that the schoolmaster was hungover and was buying himself some time to sleep off the hefeweizen from last night and that's why he assigned the students the task of adding up the whole numbers from 1 to 100. He already knew the answer and figured it would keep the students busy for a while.

Gauss reasoned that the sum of the whole numbers from 1 to 100 could be written as one long string of addition, and then rearrange that string of addition as desired based upon the commutative property of addition. Gauss took the original string of addition,
1+2+3+...+49+50+51+52+...+98+99+100

and rearranged it to look like this:
1+100+2+99+3+98+...49+52+50+51.

Adding parentheses won't matter so we add them as a visual aid:

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

It quickly becomes clear that we have 50 pairings and each pairing adds up to 101 which meant Gauss could rewrite the problem simply as 50 x 101 which is 5,050.

Gauss walked up to the schoolmaster's desk minutes later with the correct answer.

This is how Gauss's formula for finding the sum of an arithmetic sequence, aka finding the arithmetic series, came to be.

If we let n= the number of elements in the series, and let T(initial) and T(final) represent the first and last elements in the series, Gauss's formula is:

The sequence sum, aka the series = (n/2) x [T(initial) + T(final)]
In the case above it would be = (100/2) x [1 + 100]
which reduces to 50 x 101 or 5,050.

Gauss's approach to adding this orderly, sequential string of numbers could be characterized by the phrase "outside in", since he likely begain by pairing off the 1 with the 100, then the 2 with the 99, etc.

About 15 years ago, I developed a formula that is similar to Gauss's except, instead of using an outside in approach to find the sum of an arithmetic sequence, this "sibling" formula uses an inside out approach to find the product of a factorial which is like an arithmetic series that is multiplied instead of added.

I don't know what it's useful for, but, as far as I know, it is a novel contribution to the field of mathematics.
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