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Gaussian molecular formula, also called small Gauss, is a formula for the sum of the first consecutive natural numbers:
This series is a special case of arithmetic series and their sums are called triangular numbers.
Illustration
You can follow illustrate the formula: We enter the numbers from 1 to ascending in a row. Below to write the numbers in reverse order.
The sum of the columns results in each case the value Since there are columns, the sum of the numbers of both lines is equal To determine the sum of the numbers in a row, the result is halved, and it results in the above formula:
Origin of name
This sum formula as well as the empirical formula for the first n square numbers was already known in the pre-Greek mathematics.
Carl Friedrich Gauss discovered this formula as a nine year-old student again. The story is narrated by Wolfgang Sartorius von Walter Hausen: " The young Gauss had hardly entered the computing Classe when Büttner gave the summation of an arithmetic series. The task, however, was hardly pronounced as the Gauss board with the lowly spoken in Braunschweig dialect words throws on the table: " Ligget se ' " ( Because it is. ) " The exact task is not known.. It is often reported that the students had Büttner add the numbers from 1 to 100 (according to other sources from 1 to 60 ) and Gauss noted that the first and the last number (1 100 ), the second and the second to last number (2 99 ), etc. together always give 101. The value of this sum arises as to 101 times 50
According to the prevailing conditions Büttner taught about 100 students in a class. At that time punishment with the so-called Karwatsche ( leather whip ) were common. Sartorius reports: " At the end of the hour, the computing boards were then reversed; with a single number was up by Gauss and Büttner as checked the example which his own amazement of everyone present was found to be correct, while many of the rest were false and were soon rectified with the Karwatsche. " Büttner soon realized that Gauss in his class could not learn anything.
Evidence
For this sum formula, there is ample evidence. Besides the above presented evidence led to the forward - and backward - summation is still following the general principle of interest:
To demonstrate that
For all natural, it is sufficient
To show.
In fact, this applies here: and.
Also a proof of the Gaussian sum formula by induction is possible.
Related sums
Occasionally, the sum formulas for the sum of the even or the odd numbers are required:
The first formula is obtained by multiplying the basic formula with 2 The sum of the odd numbers is given by the difference of the natural and straight:
The similar-looking sum of the squares
Is called a quadratic Pyramidalzahl. A generalization to an arbitrary positive integer as the exponent is the Faulhabersche formula.