O Sistema: Números triangulares e a soma de Gauss

quinta-feira, 17 de maio de 2012

Números triangulares e a soma de Gauss

A propósito do famoso item 4.

Desculpem estar em Inglês.

"The triangular number is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. This is illustrated above for , , .... The triangular numbers are therefore 1, , , , ..., so for , 2, ..., the first few are 1, 3, 6, 10, 15, 21, ..."

Weisstein, Eric W. "Triangular Number." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/TriangularNumber.html

"The next example is one that is associated with Carl Friedrich Gauss. As one version of the story goes, when Gauss was 10 years old his teacher, Herr Büttner, asked the students to sum the integers from 1 to 100. Gauss did it almost instantly. It is believed that he did it by the following method.

Write the sum horizontally forwards and backwards as:

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

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

Now add vertically. When you do this, you will get 101 one hundred times; in other words, you get (101)(100). This is twice the sum that you needed, so the answer must be (101)(100)/2. There is nothing special about the integer 100. If you try this with a general positive integer n, you will see that 1 + 2 + 3 + ... + n = n(n+1)/2 for every positive integer n. What a nice formula! Is something like this true for the sums of squares of the first n integers? Indeed it is. We'll give it a rigorous proof using mathematical induction."

Daepp, Ulrich, and Pamela Gorkin. 2003. Reading, Writing, and Proving: A Closer Look at Mathematics. New York: Springer Science+Business Media. (p. 209.)

In Versions of the Gauss Schoolroom Anecdote, http://www.sigmaxi.org/amscionline/gauss-snippets.html