# Circles in Triangles Limit Problem

• Mar 7th 2010, 06:09 PM
Lord Darkin
Circles in Triangles Limit Problem
Let T be the equilaterall triangle with sides = 1 length.

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

$\displaystyle a_{1} = 1$

$\displaystyle a_{2} = 3$

$\displaystyle a_{3} = 6$

$\displaystyle a_{4} = 10$

Let $\displaystyle A_{n}$ be the combined area of the $\displaystyle a_{n}$ circles. Find the limit of $\displaystyle A_{n}$ 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 ... ?
• Mar 7th 2010, 06:12 PM
jass10816
Without giving it much thought, it seems like the limit should just be the area of the triangle, which is $\displaystyle \frac{1}{2}$

EDIT: Nevermind. I misread your question. A formula that might help you is that $\displaystyle a_n=\frac{n(n+1)}{2}$ Try to find a formula for $\displaystyle A_n$ For example, $\displaystyle A_1=a_1*\pi*\frac{3}{16}$
• Mar 8th 2010, 02:20 AM
Lord Darkin
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.
• Mar 8th 2010, 07:21 AM
jass10816
Yes that is the r^2. Draw a picture and it might be clearer. $\displaystyle (\frac{\sqrt{1^2-.5^2}}{2})^2=\frac{3}{16}$
• Mar 8th 2010, 01:53 PM
Lord Darkin
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.