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Math Help - Circles in Triangles Limit Problem

  1. #1
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    Circles in Triangles Limit Problem

    Let T be the equilaterall triangle with sides = 1 length.

    Let a_{n} be the # of circles that can be packed tightly in n rows inside the triangle.

     <br />
a_{1} = 1<br />

     <br />
a_{2} = 3<br />

     <br />
a_{3} = 6<br />

     <br />
a_{4} = 10<br />

    Let A_{n} be the combined area of the a_{n} circles. Find the limit of A_{n} as n -> infinity.

    I believe I have to find a series for the number of circles in the triangles, right? So I need to find something to represent 1, 3, 6, 10, 15 ... ?
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  2. #2
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    Without giving it much thought, it seems like the limit should just be the area of the triangle, which is \frac{1}{2}

    EDIT: Nevermind. I misread your question. A formula that might help you is that a_n=\frac{n(n+1)}{2} Try to find a formula for A_n For example, A_1=a_1*\pi*\frac{3}{16}
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  3. #3
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    How did you get the 3/16 part? A = pi * r^2 so I know it's r^2 but I just want to make sure I'm on the same page.
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  4. #4
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    Yes that is the r^2. Draw a picture and it might be clearer. (\frac{\sqrt{1^2-.5^2}}{2})^2=\frac{3}{16}
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  5. #5
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    When I attempt to get one circle's radius squared, I get 3/12 so I'm confused.

    Also I still don't know how to get a formula for An.
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