Re: Finding the general term of a sequence

Dear Jan:

Although you did not have time to answer the previous letter, I am in a
hurry to inform you of my solution! It turns out that I can construct a
triangle and find a general formula for the radius ri or circle Ci in
terms of the radius of the circle C(i-1). Let's presume the C1 has
point Pt in common w/ tangent line T. Now, let's draw the diagonal from
the center of the circle C to Pt. This diagonal clearly has length
sqrt(2). Moreover, the angle C-Pt-P is 45 degrees. Let's draw another
line l from the center of C1 to the center of C. This line clearly has
length 1+r1 where r1 is the radius of C1. Connecting the center of C1
to Pt we get the third line. Thus we have triangle w/ three sides (two
of which are unknown) and a known angle. Clearly, using the law of
cosines we can build an equation and solve for r. However, we can draw
line l from center of C2 when r1 is known. Thus we can solve for r2. In
general, the following sequence gives the answer for ri in terms of
r(i-1):
ri=(4r(i-1)-4(r(i-1)^2)-1)/(4r(i-1)-4). Right now I am trying to arrive
at your result.

Thank You Very Much.

Truly Yours, Simon Dexter.

.

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