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Sum of Triangular Numbers

Sum of Triangular Numbers
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2016-03-09 23:30
TriangleSequenceCalculusGeometric Construction
This example demonstrates the geometric and conceptual representation of the sum of triangular numbers. It involves core mathematical concepts such as sequences, geometric constructions, and integral calculus. The construction utilizes GeoGebra commands like Point, Intersect, Segment, Polygon, Vector, Translate, and Mirror. Suitable for high school and above, it illustrates the intersection of number theory and geometry.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Segment(A, B) → a_4
  4. Point(xAxis) → C
  5. Segment(B, C) → b
  6. Segment(C, D) → c
  7. Point(a_4) → E
  8. Segment(D, E) → d
  9. Point(a_4) → F
  10. Segment(F, G) → e
  11. Point(b) → H
  12. Segment(G, H) → f
  13. Point(b) → I
  14. Segment(I, J) → g
  15. Point(a_4) → K
  16. Segment(J, K) → h
  17. Point(b) → L
  18. Segment(L, M) → i
  19. Point(a_4) → N
  20. Segment(M, N) → j
  21. Point(b) → O
  22. Segment(O, P) → k
  23. Segment(A, P) → l
  24. Intersect(j, k) → Q
  25. Intersect(h, i) → R
  26. Intersect(e, g) → S
  27. Intersect(d, f) → T
  28. Polygon(A, P, C, B, Q, M, R, J, S, G, T, D) → poly1, a_1, d_1, c_1, b_1, p, q, m, r, j_1, s, g_1, t
  29. Segment(A_1, Z) → n
  30. Intersect(a_4, l) → B_1
  31. Segment(Z, B_1) → e_1
  32. Vector(U, V) → u
  33. Translate(A, u) → A'
  34. Translate(P, u) → P'
  35. Translate(Q, u) → Q'
  36. Translate(M, u) → M'
  37. Translate(R, u) → R'
  38. Translate(J, u) → J'
  39. Translate(S, u) → S'
  40. Translate(G, u) → G'
  41. Translate(T, u) → T'
  42. Translate(D, u) → D'
  43. Translate(C, u) → C'
  44. Translate(B, u) → B'
  45. Polygon(A', P', C', B', Q', M', R', J', S', G', T', D') → poly1', a', d', c', b', p', q', m', r', j', s', g', t'
  46. Mirror(A', Z) → A''
  47. Mirror(P', Z) → P''
  48. Mirror(Q', Z) → Q''
  49. Mirror(M', Z) → M''
  50. Mirror(R', Z) → R''
  51. Mirror(J', Z) → J''
  52. Mirror(S', Z) → S''
  53. Mirror(G', Z) → G''
  54. Mirror(T', Z) → T''
  55. Mirror(D', Z) → D''
  56. Mirror(C', Z) → C''
  57. Mirror(B', Z) → B''
  58. Polygon(A'', P'', C'', B'', Q'', M'', R'', J'', S'', G'', T'', D'') → poly1'', a'', d'', c'', b'', p'', q'', m'', r'', j'', s'', g'', t''
  59. Vector(W, V) → v
  60. Translate(A'', v) → A'''
  61. Translate(P'', v) → P'''
  62. Translate(Q'', v) → Q'''
  63. Translate(M'', v) → M'''
  64. Translate(R'', v) → R'''
  65. Translate(J'', v) → J'''
  66. Translate(S'', v) → S'''
  67. Translate(G'', v) → G'''
  68. Translate(T'', v) → T'''
  69. Translate(D'', v) → D'''
  70. Translate(C'', v) → C'''
  71. Translate(B'', v) → B'''
  72. Polygon(A''', P''', C''', B''', Q''', M''', R''', J''', S''', G''', T''', D''') → poly1''', a''', d''', c''', b''', p''', q''', m''', r''', j''', s''', g''', t'''
  73. Vector(W, V) → w
  74. Translate(A''', w) → C_1
  75. Translate(P''', w) → D_1
  76. Translate(Q''', w) → E_1
  77. Translate(M''', w) → F_1
  78. Translate(R''', w) → G_1
  79. Translate(J''', w) → H_1
  80. Translate(S''', w) → I_1
  81. Translate(G''', w) → J_1
  82. Translate(T''', w) → K_1
  83. Translate(D''', w) → L_1
  84. Translate(C''', w) → M_1
  85. Translate(B''', w) → N_1
  86. Polygon(C_1, D_1, M_1, N_1, E_1, F_1, G_1, H_1, I_1, J_1, K_1, L_1) → poly2, c_2, l_1, m_1, n_1, d_2, e_2, f_1, g_2, h_1, i_1, j_2, k_1
  87. Vector(W, V) → z
  88. Translate(C_1, z) → O_1
  89. Translate(D_1, z) → P_1
  90. Translate(E_1, z) → Q_1
  91. Translate(F_1, z) → R_1
  92. Translate(G_1, z) → S_1
  93. Translate(H_1, z) → T_1
  94. Translate(I_1, z) → U_1
  95. Translate(J_1, z) → V_1
  96. Translate(K_1, z) → W_1
  97. Translate(L_1, z) → Z_1
  98. Translate(M_1, z) → A_2
  99. Translate(N_1, z) → B_2
  100. Polygon(O_1, P_1, A_2, B_2, Q_1, R_1, S_1, T_1, U_1, V_1, W_1, Z_1) → poly2', o_1, z_1, a_2, b_2, p_1, q_1, r_1, s_1, t_1, u_1, v_1, w_1
  101. Vector(W, V) → u_2
  102. Translate(O_1, u_2) → C_2
  103. Translate(P_1, u_2) → D_2
  104. Translate(Q_1, u_2) → E_2
  105. Translate(R_1, u_2) → F_2
  106. Translate(S_1, u_2) → G_2
  107. Translate(T_1, u_2) → H_2
  108. Translate(U_1, u_2) → I_2
  109. Translate(V_1, u_2) → J_2
  110. Translate(W_1, u_2) → K_2
  111. Translate(Z_1, u_2) → L_2
  112. Translate(A_2, u_2) → M_2
  113. Translate(B_2, u_2) → N_2
  114. Polygon(C_2, D_2, M_2, N_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2) → poly2'', c_3, l_2, m_2, n_2, d_3, e_3, f_2, g_3, h_2, i_2, j_3, k_2
  115. Vector(W, V) → v_2
  116. Translate(C_2, v_2) → O_2
  117. Translate(D_2, v_2) → P_2
  118. Translate(E_2, v_2) → Q_2
  119. Translate(F_2, v_2) → R_2
  120. Translate(G_2, v_2) → S_2
  121. Translate(H_2, v_2) → T_2
  122. Translate(I_2, v_2) → U_2
  123. Translate(J_2, v_2) → V_2
  124. Translate(K_2, v_2) → W_2
  125. Translate(L_2, v_2) → Z_2
  126. Translate(M_2, v_2) → A_3
  127. Translate(N_2, v_2) → B_3
  128. Polygon(O_2, P_2, A_3, B_3, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2) → poly2''', o_2, z_2, a_3, b_3, p_2, q_2, r_2, s_2, t_2, u_3, v_3, w_2
  129. Point(Circle[(0, a), 5]) → C_3
  130. Segment((0, a), C_3) → f_3
  131. Segment((1, a - 1), Point[Circle[(0, a - 1), 4]]) → h_4
  132. Segment((1, a - 1), Point[Circle[(1, a - 1), 4]]) → h_3
  133. Segment((2, a - 2), Point[Circle[(2, a - 2), 3]]) → i_3
  134. Segment((3, a - 3), Point[Circle[(3, a - 3), 2]]) → k_3
  135. Segment((4, a - 4), Point[Circle[(4, a - 4), 1]]) → l_3
  136. Segment(C_3, I_3) → m_3
  137. Segment(I_3, H_3) → n_3
  138. Point(k_3) → K_3
  139. Segment(G_3, K_3) → r_3
  140. Segment(K_3, F_3) → s_3
  141. Point(i_3) → L_3
  142. Segment(F_3, L_3) → t_3
  143. Segment(L_3, E_3) → b_4
  144. Point(h_3) → M_3
  145. Segment(E_3, M_3) → c_4
  146. Segment(M_3, D_3) → d_4
  147. Intersect(f_3, yAxis) → N_3
  148. Segment(D_3, N_3) → e_4
  149. Segment(H_3, J_3) → p_3
  150. Segment(J_3, G_3) → q_3
  151. Point(f_3) → O_3
  152. Segment(J_3, O_3) → f_4
  153. Point(f_3) → P_3
  154. Segment(K_3, P_3) → g_4
  155. Point(f_3) → Q_3
  156. Segment(L_3, Q_3) → i_4
  157. Point(f_3) → R_3
  158. Segment(M_3, R_3) → j_4
  159. Polygon(N_3, D_3, I_3, C_3, M_3, E_3, L_3, F_3, K_3, G_3, J_3, H_3) → poly3, n_4, h_5, i_5, c_5, d_5, m_4, e_5, l_4, f_5, k_4, g_5, j_5
  160. Point(Circle[(0, a - 1), 5]) → S_3
  161. Segment((0, a - 1), S_3) → q_4
  162. Point(Circle[(1, a - 2), 4]) → T_3
  163. Segment((1, a - 2), T_3) → r_4
  164. Point(Circle[(2, a - 3), 3]) → U_3
  165. Segment((2, a - 3), U_3) → s_4
  166. Point(Circle[(3, a - 4), 2]) → V_3
  167. Segment((3, a - 4), V_3) → t_4