Mathematical induction in proving of theorems about natural numbers divisibility
Journal Title: Problemy Nauk Stosowanych - Year 2013, Vol 1, Issue
Abstract
Introduction and aims: This paper presents the concept of the division of mathematical expressions with natural variable related to the problem of divisibility. The paper shows some proofs of selected problem. The main aim of this paper is to show a few proofs of theorems about divisibility of expressions by using the method of mathematical induction. Material and methods: In this paper have been solved examples from different sources. Considered problems contain: only polynomials, the sum of powers of different bases (and constant as a component), the sum of the products of powers with different bases (and constant as a component), the sum of the powers and polynomials, the sum of the products of powers and polynomials, the sum containing the power of (-1), Fibonacci sequence, the expression containing a power of the power and problems containing power in divider. In the paper has been used the method of mathematical induction. Results: It has been shown 16 proofs of problems by using mathematical induction. In some examples have been used the additional lemmas which complete the main proof. Conclusion: Using some properties of divisibility theorems and the theorem about mathematical induction allow to show proofs which refer to the divisibility by natural number of various mathematical expressions with natural variable n.
Authors and Affiliations
Krystyna Żywuszko, Andrzej Antoni Czajkowski
Keywords
Related Articles
- EP ID EP206643
- DOI -
- Views 68
- Downloads 0
Analiza parametrów fali tętna dla noworodka podczas tachykardii - Przypadek kazuistyczny
Wstęp i cele: W pracy przedstawiono analizę saturacji, tętna i amplitudy fali tętna w pulsoksymetrii noworodków podczas występowania tachykardii. Głównym celem pracy jest analiza parametrów fali tętna dla wybranych dwóch...
Recurrence formula, differentia compound and differentia equations for Hermite polynomials
Introduction and aim: The paper presents a recurrence formula, some differential compounds and differential equation for Hermite polynomials. The aim of the discussion was to give some proofs of presented dependences. Ma...
Orthogonality of Legendre polynomials
Introduction and aim: The paper presents some Legendre polynomials, orthogonality condition for Legendre polynomials, recurrence formula and differential equation for Legendre polynomials. The aim of the discussion was t...
Główne właściwości spiralnej tomografii komputerowej
Wstęp i cel: Największy rozwój technologiczny systemów tomograficznych, nastąpił w latach 90-tych. Związany był on z wprowadzeniem pierścieni ślizgowych, dzięki którym możliwe było wykonanie pomiaru w trybie ciągłym wzdł...
Algorytm w programie Mathematica do wyznaczania parametrów krzywej pulsoksymetrii u noworodków
Wstęp i cele: W pracy przedstawiono koncepcję zdefiniowania nowych parametrów pulsoksymetrii noworodka uzyskanych z obrazu monitora. Głównym celem pracy jest opracowanie algorytmu w programie numerycznym Mathematica do u...