Email addresses Abstract The manuscript should contain an abstract. The abstract should be self-contained, citation-free, and should not exceed words. Introduction This section should be succinct, with no subheadings.

Properties of -trapezoidal numbers have been studied together with their trapezoidal representations. In the special case wherethe characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed -trapezoidal numberthe ways and the number of ways to write as an arithmetic series with common difference have been determined.

Some remarks on -trapezoidal numbers have been provided as well. Introduction A triangular number is a figurate number that can be represented by an equilateral triangular arrangement of points equally spaced. For each positive integerthe th triangular number is the number of points composing a triangle with points on a side and is equal to the sum of the natural numbers of the form The th triangular number can be represented as points in an equilateral triangle as in Figure 1.

The th triangular number. Triangular numbers have been studied since the ancient Greeks. The Pythagoreans revered the Tetractys which is. Triangular numbers have applications to other areas of number theory, such as perfect numbers and binomial coefficients.

They are also practically the simplest example of an arithmetic sequence. Therefore, the triangular numbers have fascinated people and cultures all over the world see [ 1 — 3 ] and references therein. A trapezoidal number see [ 4 ], e. Precisely, a positive integer is a trapezoidal number if for some integers and.

Trapezoidal numbers form an important class of figurate numbers that has extensively been studied see [ 2 — 7 ]. From the definition, it is not difficult to see that every trapezoidal number can be represented by a rearrangement of points in the plane as a trapezoid as in Figure 2.

For convenience, denote by the number of the form 1. The characterization and enumeration of trapezoidal numbers have been given in [ 48 ]. The main results are summarized as follows. Trapezoidal number Theorem 1 [ 4Proposition 1]. Let be a positive integer.

Then is a trapezoidal number if and only if is not of the form for all Theorem 2 [ 4Proposition 2]. Let be a positive integer such that and are distinct odd primes, and is an integer for all.

Then is a trapezoidal number and there are ways of writing as a sum of at least two consecutive integers. Some properties of nontrapezoidal numbers can be found in [ 9 ]. Triangular numbers and trapezoidal numbers have a closed connection see Section 2 for more details with a rectangular number which is defined to be where are integers.

A rectangular number can be represented as a rectangle as in Figure 3.Issue of Plagiarism in scientific writing. I think this is partly due to the fact that we have no tradition of teaching academic writing at all; although writing is essential to an academic's. ISSN: (Online) IJMMS covers the broad field of mechatronics (ashio-midori.com ashio-midori.comcience.

aerospace. computer.g. network technologies for manufacturing Not sure if . The Journal of Academic Writing is an international, peer-reviewed journal that focuses on the teaching, tutoring, researching, administration and development of academic writing in higher education in Europe.

This kind of writing assignment is meant to be interactive, as students engage with ideas and experiences that bring about questions, comparisons, insights, criticisms, speculations, and tentative conclusions. The International Journal of Medicine and Medical Sciences (IJMMS) (ISSN: ) welcomes the submission of high-quality solicited and unsolicited articles, in English, in all areas of medicine and medical ashio-midori.come medicine is a diverse field, articles should address questions utilizing a variety of methods and theoretical perspectives.

Handbook for Academic Medicine Writing Workshop.

1. This reference handbook was compiled and annotated by members of the Academic Medicine Editorial Staff. Managing Editor Anne Farmakidis. Director, Staff Editing Albert Bradford. Senior Staff Editors Mary Beth DeVilbiss Jennifer Campi.

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