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Visual Proof of the Sum of the First Odd Numbers

Visual Proof of the Sum of the First Odd Numbers
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2020-05-04 19:26
PolygonSequenceTransformationGeometric ConstructionSlider Animation
① This example demonstrates the classic identity that the sum of the first n odd numbers equals n², using a visual geometric construction that rearranges odd-numbered blocks into a square. ② The core mathematical concepts include arithmetic series, square numbers, and the geometric interpretation of algebraic identities, involving sequences, rational numbers, and series. ③ The construction uses GeoGebra commands such as Rotate, Mirror, Midpoint, Polygon, Tetrahedron, Point, Intersect, and Line to create symmetric shapes and precisely position vertices and edges, enabling a clear visual transformation. ④ It is suitable for middle and high school mathematics education, particularly for teaching summation formulas, exploring connections between geometry and algebra, and as an interactive demonstration or student exploration activity.

GeoGebra Commands & Parameters

  1. Rotate(A, 120°) → A'
  2. Rotate(A, (-(120°))) → A'_1
  3. Polygon(A, A', A'_1) → t1, a'_1, a, a'
  4. Tetrahedron(t1, true) → b, B, face1, faceAA'B, face2, arêteAB, arête1, arêteA'B
  5. Point(a'_1) → C
  6. Intersect(a'_1, yAxis) → D
  7. Point(a'_1) → E
  8. Point(a'_1) → F
  9. Point(a'_1) → G
  10. Line(G, arêteAB) → f
  11. Line(F, arêteAB) → g
  12. Line(E, arêteAB) → h
  13. Line(D, arêteAB) → i
  14. Line(C, arêteAB) → j
  15. Line(C, arêteA'B) → k
  16. Line(D, arêteA'B) → l
  17. Line(E, arêteA'B) → m
  18. Line(F, arêteA'B) → n
  19. Line(G, arêteA'B) → p
  20. Intersect(k, arêteAB) → H
  21. Intersect(f, arêteA'B) → I
  22. Intersect(arêteAB, l) → J
  23. Intersect(f, k) → K
  24. Intersect(arêteA'B, g) → L
  25. Intersect(m, arêteAB) → M
  26. Intersect(f, l) → N
  27. Intersect(k, g) → O
  28. Intersect(arêteA'B, h) → P
  29. Mirror(I, q) → I'
  30. Mirror(K, q) → K'
  31. Mirror(L, q) → L'
  32. Mirror(N, q) → N'
  33. Mirror(O, q) → O'
  34. Mirror(P, q) → P'
  35. Mirror(G, q) → G'
  36. Mirror(F, q) → F'
  37. Mirror(E, q) → E'
  38. Mirror(D, q) → D'
  39. Mirror(C, q) → C'
  40. Mirror(A', q) → A''
  41. Polygon(P', M, P) → t2, p_1, p', m_1
  42. Tetrahedron(t2, true) → c, Q, facePP'Q, faceMP'Q, faceMPQ, arêteP'Q, arêtePQ, arêteMQ
  43. Line(K', K) → r
  44. Rotate(l3, (s(d) * an), r) → l3'
  45. Rotate(l5, (s(d) * an), r) → l5'
  46. Rotate(l7, (s(d) * an), r) → l7'
  47. Rotate(c, (s(d) * an), r) → c'
  48. Rotate(B, (s(d) * an), r) → B'
  49. Midpoint(P, P') → R
  50. Rotate(R, (s(d) * an), r) → R'
  51. Midpoint(M, B) → S
  52. Rotate(S, (s(d) * an), r) → S'
  53. Line(R', S') → t
  54. Rotate(l3', (s(d) * 45°), t) → l3''
  55. Rotate(l7', (s(d) * 45°), t) → l7''
  56. Rotate(l5', (s(d) * 45°), t) → l5''
  57. Rotate(B', (s(d) * 45°), t) → B''
  58. Rotate(c', (s(d) * 45°), t) → c''
  59. Rotate(Segment[Rotate[P, (s(d) * an), r], Rotate[P', (s(d) * an), r]], (s(d) * 45°), t) → e

AI Drawing Prompt Reference

Draw an equilateral triangle and rotate it 120 degrees twice to form a larger triangle with three vertices. Place several points along one side and draw parallel lines from these points to the opposite side, creating a grid-like division. Mirror the triangle across a symmetry axis to obtain a symmetric counterpart. Within the mirrored triangle, connect corresponding points with line segments to form smaller triangles and quadrilaterals. Finally, rotate part of the figure around a line by a certain angle to show how odd-numbered blocks rearrange into a square, illustrating that the sum of the first n odd numbers equals n squared.
The prompt above was compiled from the original drawing commands and is for reference only.