Gauss aritmetic serie

25 11 2007

Last year I had Algorithms and Complexity classes and sometimes we had to make the asymptotic analysis of some algorithms. There we used many times the following summation:

\sum_{i=0}^{n} i = \frac{n(n+1)}{2}

gaussBack in then I didn’t care about what was behind that formula. Currently I am reading the Knuth book Concrete Mathematics, and there was the explanation…

In fact Gauss is who discovered this expression. There is a famous story about how Gauss discovered it:

In primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the correct answer within seconds by a flash of mathematical insight, to the astonishment of his teacher and his assistant Martin Bartels. Gauss had realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050

in wikipedia

I think that’s only a story, nothing more. Like others stories that said that Einstein was terrible in math at school.

Anyway, Gauss was really a brilliant mind in many fields and won a Copley Medal in 1838, the highest award of Royal Society of London.

If you don’t understand the definition in the previous quotation, here is the mathematical proof:


(click in the image to see it better)

I think the image is pretty clear, brilliant indeed!

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4 responses

26 11 2007
Regedor

“Gauss is who discovered this expression” bahh.. I also discovered it when had to do a program that sum the first 100 ints!! I hate that guy if was not him maybe that formula would be now called regedor’s expression lololol

26 11 2007
ulisses

you were only 223 years late…

1 12 2007
Nuno Job

Better late then never..

Já agora podias ter explicado isto melhor. Assim o artigo nao tem interesse. Se tivesses desenhado um triangulo, mostrado a area e tal tinha mais piada :P

abraço

15 09 2010
Guillermo Bautista

I have a similar article here, only a little intuitive:

http://math4allages.wordpress.com/2010/09/15/sum-first-n-positive-integers/

You may want to check it out.

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