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Investigating Triangular Numbers by Quadratic Function Along with Sequences and Set Operation

Undergraduate #68
Discipline: Mathematics and Statistics
Subcategory:

Angelique Phifer - Savannah State University


A positive integer n is a Triangular Number if and only if T_n=(n(n+1))/2 . We define a quadratic function f(x)= x^2+x-2m for any positive integer m and discuss a necessary and sufficient condition for m to be triangular, and we prove a theorem associated to a rational root d of f(x) and a triangular T_n. We also define a polynomial function P(x)=∏_(i=1)^2n▒f_i (x) given f_i (x)= x^2+x-2T_i for each triangular number T_i and prove ∏_(i=1)^2n▒R_i (x) = (-2)^n ∏_(i=1)^n▒T_i where R_i is a root of f_i (x) for each i ≥1 . Besides we derive sequences of quotients from a sequence of triangular numbers and derive a recurrence relation to prove a theorem associated to triangular numbers.

Funder Acknowledgement(s): Chellu S. Chetty, Associate VP of Research & Sponsored Programs, Savannah State University; Devi Chellu, NIH-MARC U* STAR/ RISE Program /NSF-PSLSAMP Program Manager, Savannah State University

Faculty Advisor: Tilahun Muche, muchet@savannahstate.edu

Role: My mentor introduced me to the concept of Triangular Numbers and formula. Afterwards I took two functions, a polynomial function and a quadratic function, and I proved theorems that were associated with the functions and that were also able to correlated back to the concept of Triangular Numbers. I was also able to show the relationship of a sequence of quotients and a sequence of triangular, while proving a theorem that corresponded to Triangular Numbers.

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This material is based upon work supported by the National Science Foundation (NSF) under Grant No. DUE-1930047. Any opinions, findings, interpretations, conclusions or recommendations expressed in this material are those of its authors and do not represent the views of the AAAS Board of Directors, the Council of AAAS, AAAS’ membership or the National Science Foundation.

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