**Problem 8**

In the drawing below
two circles A and B, each having a radius of 1 are tangent to each other and
also tangent to a horizontal line.
Circle C_{1} having a diameter d_{1} is inscribed within
circle A, circle B, and the horizontal tangent
line. Circle C_{2} having a
diameter d_{2} is inscribed within circles A, B, and C_{1}. Circle C_{3} having a diameter d_{3}
is inscribed within circles A, B, and C_{2}. For n = 2, 3, 4,…
circle C_{n} having a diameter d_{n} is inscribed within circles A, B, and C_{n-1}. Find d_{n}
as a function of the sum of the first n integers 1 + 2 + 3 + ∙ ∙
∙ + n.

[Problem
submitted by Vin Lee, LACC Associate Professor of
Mathematics.]

**Solution for Problem 8:**

For each circle, C_{n}, let r_{n}
be its radius. Form a right triangle
whose hypotenuse, c, is the line segment connecting the center of circle C_{1}
to the center of circle B and whose other two sides, a and b, are horizontal
and vertical respectively. Then using
the Pythagorean Theorem

_{}

_{}

_{}

_{}

_{}

Now repeat this
procedure forming a right triangle whose hypotenuse is the line segment
connecting the center of C_{2} to the center of B and whose other sides
are horizontal and vertical. For this
circle

_{} _{} _{}

For C_{3} and
C_{4} this procedure gives the following results:

_{} _{} _{}

_{} _{} _{}

Doubling these radii
gives the following sequence of diameters:

_{}

_{}

_{}

_{}

Therefore,

_{}.