Abstract / Excerpt:

Objectives: The purpose of this study is to look into the patterns and relationships of polygonal numbers. This study will attempt to discover arithmetic and geometric characteristics.
Methodology: From the reading materials, the researcher extracted all the common plane figurate numbers starting from the first order polygonal numbers to the second order polygonal numbers until the sixth order polygonal numbers.
Conclusions: Apparently, there are arithmetical patterns and relationships that exist among polygonal numbers. (1) The nth square number is the sum of the nth and the (n-1)th triangular numbers. (2) The nth square number is equal to n plus two times the (n-1)th triangular number. (3) The nth pentagonal number is equal to twice the (n-1)th triangular number plus the nth triangular number. (4) The nth pentagonal number is equal to the (n-1)th triangular number plus the nth square number. (5) The nth pentagonal number is equal to n plus three times the (n-1)th triangular number. (6) The nth hexagonal number is equal to the sum of the nth triangular number plus three times the (n-1)th triangular number. (7) The nth hexagonal number is the sum of the nth pentagonal number plus the (n-1)th triangular number. (8) The nth hexagonal number is equal to the nth square number plus twice the (n-1)th triangular number. (9) The nth hexagonal number is equal to four (n-1)th triangular number plus n. (10) The nth heptagonal number is equal to the nth triangulatr number plus n. (11) The nth heptagonal number is equal to the nth hexagonal number plus the (n-1) triangular number. (12) The nth heptagonal number is equal to the nth pentagonal number plus twice the (n-1)th triangular numbers. (13) The nth heptagonal number is equal to the nth square number plus three (n-1)th triangular numbers. (14) The nth heptagonal number is equal to five (n-1)th triangular number plus n. (15) The nth octgonal number is equal to the nth triangular number plus five (n-1)th triangular numbers. (16) The nth octagonal number is the sum of the nth heptagonal number and the (n-1)th triangular number. (17) The nth octagonal number is equal to the sum of the nth hexagonal number plus two times the (n-1)th triangular number. (18) The nth octagonal number is equal to the nth pentagonal number plus three (n-1)th triangular numbers. (19) The nth octagonal number is equal to the nth square number plus four (n-1)th triangular numbers.
(20) The nth octagonal number is equal to six (n-1)th triangular number plus n. The above conclusions suggest that any polygonal number can be separated into triangular numbers as evidenced by the presence of triangular configurations in each pattern or relationship.

Info

Source Institution	Ateneo de Davao University
Unit	Social Science
Authors	Tuazon, Maria Ana Vilela,
Page Count	1
Place of Publication	Davao City
Original Publication Date	March 1, 1993
Tags	Dissertations, Mathematics

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