## Thursday, April 06, 2006

### Conquest of the Arithmetic Progression

May June issue of The American Scientist contains an essay by Brian Hayes, where he takes us through a number of anecdotes and references describing all variations of the popular story of Carl Friedrich Gauss conquering the world of arithmetic progression as a little school kid. The essay begins with Brian's own version of the story
In the 1780s a provincial German schoolmaster gave his class the tedious assignment of summing the first 100 integers. The teacher's aim was to keep the kids quiet for half an hour, but one young pupil almost immediately produced an answer: 1 + 2 + 3 + ... + 98 + 99 + 100 = 5,050. The smart aleck was Carl Friedrich Gauss, who would go on to join the short list of candidates for greatest mathematician ever. Gauss was not a calculating prodigy who added up all those numbers in his head. He had a deeper insight: If you "fold" the series of numbers in the middle and add them in pairs—1 + 100, 2 + 99, 3 + 98, and so on—all the pairs sum to 101. There are 50 such pairs, and so the grand total is simply 50×101. The more general formula, for a list of consecutive numbers from 1 through n, is n(n + 1)/2.

And then he digs deep into the research that he did to collect all versions of the famous anecdote. A fascinating read ..