Posted by HanuMath25/11 on Sunday, November 25, 2012 at 5:23pm.
A fractal is created: A circle is drawn with radius 8 cm. Another circle is drawn with half the radius of the previous circle. The new circle is tangent to the previous circle. Suppose this pattern continues through five steps. What is the sum of the areas of the circles? Express your answer as an exact fraction.
9840 = 3(1-3^n)/(1-3)
n = 8
a+ar+ar^2 = 35
a*ar*ar^2 = 1000
a(1+r+r^2) = 35
a^3 r^3 = 1000
by inspection, since 1000 = 2^3 * 5^3, a and r are 2 and 5.
the only factors of 35 are 5 and 7, so
a=5
r=2
(a,b,c) = (5,10,20)
if only one circle of each radius is drawn, the radii are
8,4,2,1,1/2,1/4
pi(8^2 + 4^2 + 2^2 + 1^2 + (1/2)^2 + (1/4)^2) = 1365/16 pi
Note that since
sum(1,infinity) 1/n^2 = pi^2/6, if the sequence were carried out forever, the area would be
pi * 64 * pi^2/6
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