Multidimensional Generalized Arithmetic Progressions of Multiplicity One

This article is a review work to expose the extended concepts of arithmetic progressions originating by applying the concepts of an arbitrary dimension and a fixed multiplicity (one but can be expanded for more than one also) applied on different common differences, which was published in chapters seven and eight in two books on multidimensional arithmetic progressions cited in the references and in the article. In this article we will report the advanced arithmetic progressions with multiplicity one of different dimensions from one to r and will discuss the formula to find their general terms and the sums of first n terms. We have left the discussion on the formulae to find the arithmetic means between any two arbitrary members of such generalized progressions so that mathematics teachers and learners can find it for useful teaching with a new look and a research oriented approach. Thus the article opens many new areas of research and its applications.