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Math Help - circles, hard

  1. #1
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    circles, hard

    hey guys, hope you can help me solve this problem

    One hundred concentric circles with radii 1, 2, 3, , 100 are drawn in a plane. The interior of the circle of radius one is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color.
    What is the ratio of the total area of the green regions to the area of the largest circle?
    thanks
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  2. #2
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    Hello, anime_mania!

    It seems to be fairly straight-forward . . .


    One hundred concentric circles with radii 1, 2, 3, , 100 are drawn in a plane. The interior
    of the circle of radius one is colored red, and each region bounded by consecutive circles
    is colored either red or green, with no two adjacent regions the same color.
    What is the ratio of the total area of the green regions to the area of the largest circle?

    \begin{array}{cccc}<br />
\text{The 1st green ring has area:} & 2^2\pi-1^2\pi & = & 3\pi \\<br />
\text{The 2nd green ring has area:} & 4^2\pi-3^2\pi & = & 7\pi \\<br />
\text{The 3rd green ring has area:} & 6^2\pi-5^2\pi & = & 11\pi\\<br />
\vdots & \vdots & & \vdots \\<br />
\text{The 50th green ring has area:} & 100^2\pi - 99^2\pi &=&199\pi<br />
\end{array}


    The total area of the green regions is: . (3 + 7 + 11 + 15 + \hdots + 199)\pi

    We have an arithmetic series with first term a = 3,
    . . last term l = 199 and n = 50 terms.
    Its sum is: . \frac{50}{2}(3 + 199) \:=\:5050

    Hence, the total area of the green regions is: . 5,050\pi

    The area of the largest circle is: . 100^2\pi\:=\:10,000\pi


    Therefore, the ratio is: . \frac{5,050\pi}{10,000\pi} \;=\;\boxed{\frac{101}{200}}

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  3. #3
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    oh oh i see, thanks
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