Keywords:-
Article Content:-
Abstract
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.
References:-
References
Dawson, R. (2012). Arithmetic Polygons, The American Mathematical Monthly, 119 (8), 695 – 698.
James & James (2001). Mathematics Dictionary, 4th Edition, CBS Pub.& Dist., New Delhi.
Katz, V. J. (2019). A History of Mathematics, 3rd Edition, Pearson India Education Services Pvt. Ltd.
Sarkar, S. K. (2003). A Textbook of Discrete Mathematics, S. Chand, 123.
Ulas, M. (2005). A note on arithmetic progressions on quartic elliptic curves. Journal of Integer Sequences, 8(3), 1–5.
Wikipedia contributors. (2026, February 22). Arithmetic progression. In Wikipedia, The Free Encyclopedia. Retrieved 13:46, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Arithmetic_progression&oldid=1339910185.
Wikipedia contributors. (2025, July 19). Convergent series. In Wikipedia, The Free Encyclopedia. Retrieved 13:42, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Convergent_series&oldid=1301349366.
Wikipedia contributors. (2026, January 28). Dimension. In Wikipedia, The Free Encyclopedia. Retrieved 16:44, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Dimension&oldid=1335232290.
Wikipedia contributors. (2024, November 20). Generalized arithmetic progression. In Wikipedia, The Free Encyclopedia. Retrieved 16:59, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Generalized_arithmetic_progression&oldid=1258512328.
Wikipedia contributors. (2025, August 17). Multiplicity (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 16:40, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Multiplicity_(mathematics)&oldid=1306373388.
Wikipedia contributors. (2025, December 25). Progression. In Wikipedia, The Free Encyclopedia. Retrieved 13:33, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Progression&oldid=1329337695.
Wikipedia contributors. (2026, February 13). Sequence. In Wikipedia, The Free Encyclopedia. Retrieved 08:57, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Sequence&oldid=1338181002.
Wikipedia contributors. (2026, February 1). Series (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 13:38, February 25, 2026, from https://en.wikipedia.org/w/index.php?title=Series_(mathematics)&oldid=1336008352
Yadav, D. K. (2008a). General study on Two-Dimensional Generalized Arithmetic Progression Acta Ciencia Indica Mathematics, 34 M (2), 557 – 566.
Yadav, D. K. (2008b). General study on Two-Dimensional Generalized Arithmetic Progression with multiplicity two, Acta Ciencia Indica Mathematics, 34 M (2), 905 - 912.
Yadav, D. K. (2010). Three-Dimensional Generalized Arithmetic Progression with multiplicity one, Acta Ciencia Indica Mathematics, 36 M (1), 49 - 53.
Yadav, D. K., Chaudhary, M. K. & Lal, R. R. (2026). Three Dimensional Generalized Arithmetic Progression with Multiplicity Two, International Journal of Mathematics and Statistics Invention, 14 (2), March – April, 1 - 10.
Yadav, D. K., Lal, R. R. & Chaudhary, M. K. (2026). Three Dimensional Generalized Arithmetic Progression with Multiplicity Three, International Journal of Mathematics and Computer Research, 14 (4), April 2025, 6306 - 6315.
Yadav, D. K. (2020). Multi Dimensional Arithmetic Progression, International Edition, GRIN Verlag Publishing, Germany.
Yadav, D. K. (2026). Multidimensional Arithmetic Progressions, Indian Edition, Notion Press, India.