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 C1 having a diameter d1 is inscribed within circle A, circle B, and the horizontal tangent line.  Circle C2 having a diameter d2 is inscribed within circles A, B, and C1.  Circle C3 having a diameter d3 is inscribed within circles A, B, and C2.  For n = 2, 3, 4,… circle Cn having a diameter dn is inscribed within circles A, B, and Cn-1.  Find dn 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, Cn, let rn be its radius.  Form a right triangle whose hypotenuse, c, is the line segment connecting the center of circle C1 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 C2 to the center of B and whose other sides are horizontal and vertical.  For this circle

For C3 and C4 this procedure gives the following results:

Doubling these radii gives the following sequence of diameters:

Therefore,

.