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• Arithmetic progressions of higher order

  225-239

  Vlastimil Dlab

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  9

  The aim of this article is to clarify the role of arithmetic progressions of higher order in the set of all progressions. It is important to perceive them as the pairs of progressions closely connected by simple relations of differential or cumulative progressions, i.e. by operations denoted in the text by r and s. This duality affords in a natural way the concept of an alternating arithmetic progression that deserves further studies. All these progressions can be identified with polynomials and very special, explicitly described, recursive progressions. The results mentioned here point to a very close relationship among a series of mathematical objects and to the importance of combinatorial numbers; they are presented in a form accessible to the graduates of secondary schools.

• The sum of the same powers of the first n positive integers and the Bernoulli numbers

  91-105

  István Molnár

  Views:

  9

  The first part of this paper presents a method to calculate the sum of the same powers of the first n positive integers which is non-recursive and easy to express algorithmically. The application is demonstrated through several problems, for example by calculating the sum of arithmetic progression of degree p. The second part of the paper shows that the discussed procedure can also be used to calculate the Bernoulli numbers, and then, with the help of a known theorem, a link is established between the sum of the same powers of the first n positive integers and the Bernoulli numbers.