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三角数求和 by Earl Samuelson

三角数求和 by Earl Samuelson
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2016-03-09 23:30
测试改名0数列微积分几何作图
①本范例演示了三角数的求和图形与现象；②涉及的核心数学概念包括数列、几何构造、微积分中的积分概念；③作图要点——从下方『构造摘要』提炼用到了 Point, Intersect, Segment, Polygon, Vector, Translate, Mirror 等GeoGebra命令/对象；④适合高中及以上数学学习者，用于理解数列求和与几何构造的结合。

GeoGebra 命令及参数

  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