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Wordless Proof: Sum of Odd Numbers

Wordless Proof: Sum of Odd Numbers
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2020-03-24 04:00
PolygonFunctionSequenceGeometric Construction
This example demonstrates visually how the sum of consecutive odd numbers forms a perfect square through dynamic geometry. Key math concepts include sequences, geometric constructions, and algebraic proofs. GeoGebra commands used are Sequence, Vector, Join, Polygon, Element, and Midpoint. Suitable for middle school students and above learning sequence summation, geometric proofs, and mathematical visualization.

GeoGebra Commands & Parameters

  1. Sequence(Sequence[(i, j), i, 0, n - 1 - j], j, 0, n - 1) → list1
  2. Sequence(Sequence[(n + 1 - j + i, j - 1), i, 0, n - 1 - j], j, 1, n - 1) → list2
  3. Vector((-1, 1)) → u
  4. Join(Sequence[Sequence[Translate[(n + 1 - j + i, j - 1), Vector[((k * u) * (i + 1))]], i, 0, n - 1 - j], j, 1, n - 1]) → list3
  5. Sequence(Polygon[(0, n - 1), (0, n - 1 - i), 4], i, 1, n - 1) → listPoly
  6. Sequence(Sequence[(i, j), i, 0, n - 1 - j], j, 0, n - 1, 2) → l1
  7. Sequence(Sequence[Translate[(n + 1 - j + i, j - 1), Vector[((k * u) * (i + 1))]], i, 0, n - 1 - j], j, 1, n - 1, 2) → l3
  8. Sequence(Segment[(0, i), ((2 * n) - 2 - (2 * i), i)], i, 0, n - 2) → listSegment
  9. Sequence(Segment[(0, i), ((2 * n) - (n + 1) - i, i)], i, 0, n - 2) → l2
  10. Sequence(((2 * n) - (n + 1) - i, i), i, 0, n - 2) → l5
  11. Join(Sequence[Sequence[Translate[(n + 1 - j + i, j - 1), Vector[((k * u) * (i + 1))]], i, 0, n - 1 - j], j, 1, n - 1, 1]) → l4
  12. Vector(Element[l5, 1], Element[list3, 1]) → v
  13. Sequence(Translate[Element[l5, i], Vector[((n - i) * v)]], i, 1, Length[l5]) → l6
  14. Sequence(Segment[Element[l5, i], Element[l6, i]], i, 1, Length[l5]) → l7
  15. Polygon((0, n - 1), (0, 0), 4) → poly1
  16. Sequence(Text[(2 * k) + 1, (-1.1, n - k - 1.1)], k, 0, n - 1) → l8
  17. Sequence(Text[(2 * k) + 1 + "+"], k, 0, n - 1) → l9
  18. Sequence(A + ((1.15 * i), 0), i, 0, n) → l10
  19. Sequence(Text[Element[l8, k] + " + ", Element[l10, k]], k, 1, n) → l11
  20. Element(l5, 1) → MM
  21. Element(l6, 1) → NN
  22. Midpoint(B, C) → D
  23. Vector(D, B) → w
  24. Vector(D, C) → a