the (unofficial) Arithmetic Sequence Crash-Course

Quick links: http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-sum-of-arithmetic-progression http://www.artofproblemsolving.com/Wiki/index.php/Arithmetic_sequence _ The story of Gauss: In 1784 after his seventh birthday, child prodigy Carl Friedrich Gauss entered the public school where elementary subjects were taught and was assigned a man named Büttner as his teacher. The room in which Büttner taught was a drag school-room with a worn, uneven floor. There were about a hundred students in the class and Büttner would go back and forth, back and forth with a rod in his hand that was accepted by everyone as the final argument of the teacher. And Büttner was not afraid of arguing, either. In this school the young Gauss remained for two years without incident. Soon he had graduated to the arithmetic class in which most boys remained until their fifteenth year. It was in this class that occurred an incident which Gauss would later relate in old age with great amusement. In this class, Büttner would often hand out long, arduous calculations and then sit back in his chair and wait while the students scratched out an answer on their slates. The first student to finish would place his slate in the middle of a large table. On top of this the second placed his slate, and so on. Now the young Gauss had just entered the class when Büttner gave out for a problem [the summing of the first 100 integers]. Büttner barely had time to state the problem before Gauss threw his slate on the table, proclaiming "There it lies." Büttner was surprised, but decided to wait until the end of class to check Gaus’s slate and humiliate him. Meanwhile the other pupils continued multiplying and adding, and Büttner, with conscious dignity, walked back and forth, occasionally throwing a pitying glance toward the young Gauss. With his task ended, the boy sat quietly, not doubting for a second that his answer was correct. At the end of the hour all the students turned in their slates and the teacher read them one by one. 5047, 4050, 5100… none of them were right. In the end, the only slate left was that of the young Gauss. Büttner slowly and deliberately turned over the slate and read out loud the figure written on it…. 5050. It was the correct answer.