無窮等比級數示例
Author: 藍邦偉  老師

c=%v

$B$

a=%v

$ 無窮等比級數示例 $

$A$

$C$

Reset

k = 4 (slider from 1 to 7, step 1)
B = (0.04858913072508092, -0.027021455817588552)
h = 3 (slider from 1 to 10, step 1)
A: B + (0, k)
C: B - (h, 0)
a = Segment(B, C)
b = Segment(A, C)
c = Segment(A, B)
r: a / (a + c)
n = 1 (slider from 1 to 30, step 1)
poly_1 = Polygon(Dilate[A, r^(n), C], Dilate[B, r^(n), C], 4)
List_1 = Sequence(Polygon[Dilate[A, r^(i), C], Dilate[B, r^(i), C], 4], i, 1, n - 1)
textc: "$ a=" + (LaTeX[a]) + " $"
texta: "$ c=" + (LaTeX[c]) + " $"
a_1: a * c / (a + c)
List_x = Sequence((k, 0), k, floor(x(Corner[1])), 0)
List_y = Sequence((0, k), k, 0, floor(y(Corner[3])))
t_1: "$由相似三角形知：$
$ \frac{" + (LaTeX[c]) + "}{" + (LaTeX[a]) + "} ＝\frac{" + (LaTeX[c]) + "-x_1}{x_1}$"
t_6: "$所有正方形的面積和為：$
$ S=\frac{a_1}{1-r} =\frac{" + (LaTeX[FractionText[a_1^(2)]]) + "}{1-" + (LaTeX[FractionText[r^(2)]]) + "}=" + (LaTeX[FractionText[a_1^(2) / (1 - r^(2))]]) + "$"
t_3: "$ \frac{" + (LaTeX[c]) + "}{" + (LaTeX[a]) + "}＝\frac{" + (LaTeX[c]) + "-x_1-x_2}{x_2} =\frac{" + (LaTeX[FractionText[4 - a_1]]) + "-x_2}{x_2}$"
a_2: a_1 * r
Step = 0 (slider from 0 to 6, step 1)
t_2: "$第一個正方形的邊長與面積分別為：\;x_1=" + (LaTeX[FractionText[a * c / (a + c)]]) + " ⇒a_1=x_1^2=" + (LaTeX[FractionText[(a * c / (a + c))^(2)]]) + "$"
t_4: "$ 第二個正方形的邊長與面積為： \;x_2=" + (LaTeX[FractionText[a_2]]) + "⇒a_2=x_2^2=" + (LaTeX[FractionText[a_2^(2)]]) + "$"
t_5: "$ 正方形面積的公比r=\frac{a_2}{a_1}=" + (LaTeX[FractionText[r^(2)]]) + " $"