Invention of Arithmetic Progression

Arithmetic Progression was invented by Johann Carl Friedrich Gauss. Here is the story how he found the method.

One day at school, Gauss’s teacher wanted to take a rest and asked the students to sum the integers from 1 to 100 as busy work. To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away.His classmates and teacher were astonished, and Gauss ended up being the only pupil to calculate the correct answer.

Here is how he calculated :

First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:

1 + 2 + 3 + 4 + . . . + 99 + 100
100 + 99 + . . . + 4 + 3 + 2 + 1 By adding vertically, each pair of numbers adds up to 101:

1	+	2	+	3	+	. . .	+	98	+	99	+	100
100	+	99	+	98	+	. . .	+	3	+	2	+	1

---

101

+

101

+

101

+

. . .

+

101

+

101

+

101

Since there are 100 of these sums of 101, the total is 100101 = 10,100. Because this sum 10,100 is twice the sum of the numbers 1 through 100, we have:

1 + 2 + 3 + . . . + 98 + 99 + 100 = 100101 / 2 = 5050.  

Using Gauss´ approach, we can easily derived the formula of Arithmetic Progression.

References:
http://blog.brilliant.org/2013/03/03/arithmetic-progression-and-young-gauss-the-prince-of-mathematics/
http://www.education2000.com/demo/demo/botchtml/arithser.htm
http://www.jimloy.com/algebra/gauss.htm