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Geomatric series

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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.

  • Geomatric series - ,

    area = π(8)^2 + π(4)^2 + π(2)^2 + π(1)^2 + π(.5)^2
    = 64π + 16π + 4π + π + (1/4)π

    this is a GS where a = 64π and r = 1/4

    but , rather than using the formula in this case it is just as easy to just add them up
    total = 85.25π

    check:
    area = 64π( 1 - (1/4)^5)/(3/4) = 85.25π

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