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Relation between Pentagonal and Triangular Numbers

Relation between Pentagonal and Triangular Numbers
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2020-03-30 21:09
TrianglePolygonSequenceTransformationGeometric Construction
① This GeoGebra example dynamically illustrates the algebraic relationship between pentagonal numbers and triangular numbers, showing how the nth pentagonal number can be expressed as a combination of triangular numbers. ② Core mathematical concepts include polygonal numbers (pentagonal and triangular), integer sequences, sums of arithmetic progressions, and geometric constructions involving equilateral triangles and pentagons via translation and tiling. ③ Construction highlights: using GeoGebra commands such as Sequence, Join, Take, Translate, Element, Polygon, Vector, Line, Intersect, and Midpoint, the applet builds triangular grids and pentagonal arrays, translates triangular units to form pentagons, and displays the numerical relations with text objects. ④ Suitable for upper secondary or early university students, this example supports exploration of number theory through geometric intuition and can be used in math labs or inquiry-based learning.

GeoGebra Commands & Parameters

  1. Sequence(Vector[C, C + (cos((i * (360°) / 5)), sin((i * (360°) / 5)))], i, 0, 5 - 1) → list3b
  2. Sequence(Vector[Sum[Take[list3b, 1, i]]], i, 1, 5) → list3c
  3. Join({{C + (0, 0)}, Sequence[Sequence[Sequence[C + (0, 0) + (i * Element[list3c, t]) + (j * Element[list3b, t + 1]), j, 0, i], i, 1, n - 1], t, 1, 3]}) → list3
  4. Sequence(Polygon[C + (0, 0), C + (i, 0), 5], i, 1, n - 1) → l4
  5. Sequence(Vector[C, C + (cos((i * (360°) / 3)), sin((i * (360°) / 3)))], i, 0, 3 - 1) → l2b
  6. Sequence(Vector[Sum[Take[l2b, 1, i]]], i, 1, 3) → l2c
  7. Take(list3, Length[list3] - 2, Length[list3] - 1) → l2
  8. Take(list3, Length[list3] - 1, Length[list3]) → l5
  9. Translate(C, Vector[((n - 1) * Element[list3b, 1]) + ((n - 1) * Element[list3b, 2])]) → C'
  10. Join(list3) → l6
  11. Join(Join[Sequence[Sequence[Sequence[C + (0, 0) + (i * Element[list3c, t]) + (j * Element[list3b, t + 1]), j, 0, i], i, 1, n - 1], t, 1, 3]]) → l7
  12. Element(l7, 2) → D
  13. Polygon(C', D, E) → t2, e_1, c', d
  14. Element(l7, n^(2)) → H
  15. Polygon(J, H, I) → t1, i, j, h
  16. Sequence(Polygon[C + (0, 0), C + (i, 0), 5], i, n - 1, n - 1) → liste3
  17. Take(list3, 1, Length[list3] - 2) → l3
  18. Element(l7, Length[l7]) → K
  19. Vector(Element[l7, Length[l7]], Element[l7, Length[l7] - 1]) → u
  20. Line(L, h) → f
  21. Line(K, C) → g
  22. Intersect(f, g) → M
  23. Polygon(L, K, M) → t3, m_1, l_1, k
  24. Sequence(Translate[C, Vector[(i * Element[list3b, 1])]], i, 0, n - 1) → l8
  25. Midpoint(A_1, N) → O
  26. Vector(O, N) → v
  27. Vector(O, A_1) → w
  28. Sequence(Vector[A, A + (cos((i * (360°) / 3)), sin((i * (360°) / 3)))], i, 0, 3 - 1) → list1b
  29. Sequence(Vector[Sum[Take[list1b, 1, i]]], i, 1, 3) → list1c
  30. Sequence(Polygon[A + (0, 0), A + (i, 0), 3], i, n - 1, n - 1) → l10
  31. Join({{A + (0, 0)}, Sequence[Sequence[Sequence[A + (0, 0) + (i * Element[list1c, t]) + (j * Element[list1b, t + 1]), j, 0, i], i, 1, n - 1], t, 1, 1]}) → l9
  32. Translate(l7, Vector[C, P]) → l7'
  33. Join({{A + (0, 0)}, Sequence[Sequence[Sequence[A + (0, 0) + (i * Element[list1c, t]) + (j * Element[list1b, t + 1]), j, 0, i], i, 1, n - 2], t, 1, 1]}) → l9b
  34. Sequence(Polygon[A + (0, 0), A + (i, 0), 3], i, n - 2, n - 2) → l11
  35. Translate(liste3, Vector[C, P]) → list3'
  36. Vector(Q, R) → a
  37. Vector(S, T) → b
  38. Vector(U, V) → c
  39. Sequence(Polygon[A + (0, 0), A + (i, 0), 3], i, n - 2, n - 2) → l12

AI Drawing Prompt Reference

Create a dynamic geometric diagram illustrating the algebraic relationship between pentagonal and triangular numbers. The figure includes a pentagonal array formed by translating and tiling equilateral triangular units, alongside a triangular grid. Multiple polygons, vectors, and intersection points are shown, with a symmetric layout that visually compares the recursive relationship between pentagonal and triangular numbers.
The prompt above was compiled from the original drawing commands and is for reference only.