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A geometric approach to the general term of reductional square-triangular number sequence

Project summary

Numbers written as the sum of consecutive natural numbers starting from 1 are called triangular numbers. Triangular numbers, when modeled as points, take the form of an isosceles triangle in the plane. Numbers that are equal to the square of a natural number are called square numbers. Square numbers take the form of a square when modeled as points. Numbers that are both square and triangular are called square-triangular numbers. Square-triangular numbers can be geometrically modeled as both a square and an isosceles triangle. In this project, it is intended to determine the relationship between the terms of the square-triangular number sequence and the general term by means of geometric modeling. If an isosceles triangle with an equal number of points can be obtained from a square representing a square number, that number becomes a square-triangular number. Drawings were made for the square-triangular numbers 36 and 1225 on dotted paper. An isosceles right triangle is cut from the upper right corner of the square representing the square number. The same triangles were cut out of the cut triangle. An isosceles right triangle representing the triangular number was obtained by placing these triangles at the top left and bottom right of the remaining part of the square. Thus, it was observed geometrically that these numbers are square-triangular numbers. By generalizing this geometric approach, a new relation is found for the general term of the square-triangular reduction number sequence. A term of STn is found if S_(K_n ) = T_(Ü_n ) to represent S_(K_n ) square, T_(Ü_n ) triangular and STn square-triangular number sequences. For ST1, K1 = 1 and Ü1 =1, for n ∈ N+ , Kn+1 = 3.Kn + 2.Ün + 1 , Ün+1 = 4.Kn + 3.Ün + 1 and the general term of the reduction square-triangular number sequence, 〖ST〗_(n+1)=S_(K_(n+1) )=T_(Ü_(n+1) ) )=〖(3K_n+2Ü_n+1)〗^2. Keywords: square number, triangular number, square-triangular number, reduced sequence, general term

Students

ELİF AZRA BİLEK

Guidance counsellors

MUHAMMET ÖZDEMİR

Institution

Samsun Özel Bil Fen Lisesi
  Samsung -
  Turkey

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Gabriel Henrique Dahmer
Gabriel Henrique Dahmer
6 months ago

Congratulations!!

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Students

ELİF AZRA BİLEK

Guidance counsellors

MUHAMMET ÖZDEMİR

Institution

Samsun Özel Bil Fen Lisesi
  Samsun –
  Turkey

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