Keywords:-
Article Content:-
Abstract
This paper is a review article to expose the extended advanced concepts of geometric progressions made by applying the basic concepts of an arbitrary dimension and a fixed multiplicity (one which can be expanded for more than one also) applied on different common ratios, which was earlier published in chapter seven and eight in two books on multidimensional geometric progressions cited in the references. In this article we will report the advanced geometric progressions with multiplicity one of different dimensions from one to r and will discuss the formulae to find their general terms and the sums of first n terms. The formulae for infinite number of terms for different dimensions and multiplicities have also been discussed. We have left the discussion on the formulae to find the geometric means between any two arbitrary terms of such generalized geometric progressions so that mathematics teachers and learners can find it useful in teaching with a new look and a research-oriented approach. The article also opens many new areas of research and its applications.
References:-
References
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.
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. (2026, February 23). Geometric progression. In Wikipedia, The Free Encyclopedia. Retrieved 06:24, April 28, 2026, from https://en.wikipedia.org/w/index.php?title=Geometric_progression&oldid=1340094071.
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. (2020a). Multi-Dimensional Arithmetic Progression, International Edition, GRIN Verlag Publishing, Germany.
Yadav, D. K. (2020b). Multi-Dimensional Geometric Progression, International Edition, GRIN Verlag Publishing, Germany.
Yadav, D. K., Chaudhary, M. K. & Lal, R. R. (2026a). 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. (2026b). Three-Dimensional Generalized Arithmetic Progression with Multiplicity Three, International Journal of Mathematics and Computer Research, 14 (4), April 2026, 6306 - 6315.
Yadav, D. K. (2026a). Multidimensional Arithmetic Progressions, Indian Edition, Notion Press, India.
Yadav, D. K. (2026b). Multidimensional Geometric Progressions, Indian Edition, Notion Press, India.
Yadav, D. K. (2026c). Multidimensional Generalized Arithmetic Progression of Multiplicity One, International Journal of Mathematics and Computer Research, 14 (4), April 2026, 6324 - 6330.