**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.