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Sequence of remaining area af a triangle, when subtracting a sequence or part of a triangle repeatedly

ÀÛ¼ºÀÚ Uploader : fatman ÀÛ¼ºÀÏ Upload Date: 2019-08-18º¯°æÀÏ Update Date: 2023-07-10Á¶È¸¼ö View : 343

As shown in the figure, remove the equilateral triangle inscribed into the equilateral triangle of each side length L, and remove the equilateral triangles inscribed into the remaining equilateral triangles. Find the remaining area when repeating n times.

Area A0 of ​​equilateral triangles of length L on one side

A0 = L^2*3^(1/2)/4

n=1, A1 = A0*(3/4) = (L^2*3^(1/2)/4)*(3/4)
n=2, A2 = A1*(3/4) = (L^2*3^(1/2)/4)*(3/4)^2
n=3, A3 = A2*(3/4) = (L^2*3^(1/2)/4)*(3/4)^3
¡¦
n=n, An = An-1*(3/4) = (L^2*3^(1/2)/4)*(3/4)^(n)

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An = (L^2*3^(1/2)/4)*(3/4)^(n)
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