Obtaining the elements of Pascal’s Triangle by a new recurrence relation
DOI:
https://doi.org/10.5902/2179460X74096Keywords:
Pascal's Triangle, Arithmetic triangle, Recurrence, Difference equationsAbstract
Pascal’s Triangle is a numerical arrangement, constructed from binomial numbers and although istaught only in High School, several possible applications can already be seen in Elementary School. Theanalysis and development of these arrangements foster the student’s skills that involve critical andlogical-mathematical reasoning, in line with the provisions of Brazil’s National Common Curricular Base(BNCC). In this sense, this article promoted a detailed study of Pascal’s Triangle, indicating, in addition tothe already known tools, a new recurrence relation which permits obtaining elements of the triangleonly with elementary operations, without the use of combinatorial analysis. This new relation was alsoinserted in the context of the difference equations, being a homogeneous linear differential equationthat has the combination formula itself as a solution. Despite the recursive equation having the limitation of depending on the preceding element to the one to be calculated, its usage offers countlesspossibilities in education, and in this sense, some applications were evidenced.
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References
Affonso, A. (2014). O triˆangulo de Pascal e o Binˆomio de Newton. (mestrado profissional em matem´atica em rede nacional), UFF, Niter´oi, RJ, Brasil.
Bloch, E. D. (2011). The Real Numbers and Real Analysis. Springer.
Boyer, C. B. (1996). Hist´oria da matem´atica. Editora Blucher, 3 edition.
Brasil (2002). PCN+ Ensino M´edio: orientac¸ ˜oes educacionais complementares aos Parˆametros Curriculares Nacionais. Ciˆencias da Natureza, Matem´atica e suas Tecnologias. Bras´ılia.
Brasil (2018). Base Nacional Comum Curricular. Bras´ılia: Minist´erio da Educac¸ ˜ao.
Cutland, N. (1980). Computability: an introduction to recursive function theory. Cambridge University Press.
Davis, T. (2010). Exploring pascal’s triangle. In Mathematical Circles Topics.
Elaydi, S. (2005). Introduction to Difference Equations. Springer.
Fossa, J. A. (2017). Aleae interruptar: uma curiosa aplicac¸ ˜ao do triˆangulo de pascal. Boletim Cearense de Educac¸ ˜ao e Hist´oria da Matem´atica, 4(11):22–34.
Lima, E. L., Carvalho, P. C. P., Wagner, E., & Morgado, A. C. (2004). A Matem´atica do Ensino M´edio volume 2. Sociedade Brasileira de Matem´atica.
Lins, R. C. & Gimenez, J. (1997). Perspectivas em Aritm´etica e ´Álgebra para o s´eculo XXI. Papirus.
Lopes, M. S. & Carneiro, R. S. (2020). Triˆangulo de pascal: breve hist´oria e uma proposta did´atica para o ensino. Revista Eletrˆonica Matem´atica e Estat´ıstica em Foco, 7(1):75–97.
Marques, F. S. (2022). Recursividade em pr´aticas educativas investigativas: significados produzidos por participantes de um processo de formaç˜ao de professores de matem´atica. (programa de p´os-graduação em educação em ciˆencias e matem´atica), IFES, Vila Velha, ES, Brasil.
Morgado, A. C., Carvalho, J. B. P., & Fernandez, P. (2020). A´alise combinat´oria e probabilidade. Editora SBM, 11 edition.
Rosadas, V. D. S. (2016). Triˆangulo de Pascal: curiosidades e aplicac¸ ˜oes na escola b´asica. (mestrado em matem´atica), PUC-RJ, Rio de Janeiro, RJ, Brasil.
Santiago, T. P. (2016). Triˆangulo de Pascal: aplicac¸ ˜oes no ensino fundamental e m´edio. (mestrado profissional em matem´atica em rede nacional), UFBA, Salvador, BA, Brasil.
Santos, N. L. P. (2017). O misterioso e enigm´atico mundo de Pascal e Fibonacci. (mestrado profissional em matem´atica em rede nacional), UNESP, S˜ao Jos´e do Rio Preto, SP, Brasil.
Silva, M. O. (2015). Do triˆangulo `a pirˆamide de Pascal. (mestrado profissional em matem´atica em rede nacional), UESC, Ilh´eus, BA, Brasil.
Snustad, D. P. & Simmons, M. J. (2008). Fundamentos de Gen´etica. Guanabara, 4 edition.
Souza, C. M. (2021). Simetrias no Triˆangulo de Pascal. (mestrado profissional em matem´atica em rede nacional), UFSJ, S˜ao Jo˜ao Del Rei, MG, Brasil.
Souza, C. M. (2022). Triˆangulo de Pascal e An´alise Combinat´oria: observando padr˜oes e fazendo conjecturas. (mestrado profissional em matem´atica em rede nacional), UFF, Niter´oi, RJ, Brasil.
Souza, C. R. (2019). Os livros did´aticos de Matem´atica, a variedade de problemas propostos e o Binˆomio de Newton. (mestrado profissional em matem´atica em rede nacional), UTFPR, Pato Branco, PR, Brasil.
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