Finding an expression for the diameter of a circle. The figure shows two circles of radius $1$ that touch at $P$. 1]1 $T$ is a common tangent line; $C_1$ is the circle that touches $C$, $D$, and $T$; $C_2$ is the circle that touches $C$, $D$, and $C_1$; $C_3$ is the circle that touches $C$, $D$, and $C_2$. This procedure can be continued indefinitely and produces an infinite sequence of circles $\{C_n\}$. Find an expression for the diameter of $C_n$.

Notice, the radius say $r_1$of circle $C_1$ is given by the formula (see derivation) $$\color{red}{r_1=\frac{ab}{(\sqrt a+\sqrt b)^2}}$$ setting radii $a=b=1$ of larger identical circles, we get  $$r_1=\frac{1\cdot1}{(1+1)^2}=\frac{1}{4}$$
Now, the radius $r_2$ of circle $C_2$ externally touching the circles $C$, $D$ & $C_1$ is given by the standard formula (see derivation) 
$$\color{red}{r_2=\frac{abc}{2\sqrt{abc(a+b+c)}+(ab+bc+ca)}}$$ Setting the values of $a=1, b=1, c=r_1=\frac{1}{4}$, we get radius of circle $C_2$ as follows $$r_2=\frac{1\cdot1\cdot\frac{1}{4}}{2\sqrt{1\cdot1\cdot\frac{1}{4}\left(1+1+\frac{1}{4}\right)}+1\cdot1+1\cdot \frac{1}{4}+\frac{1}{4}\cdot 1}=\frac{1}{12}$$
Similarly, 
the radius $r_3$ of circle $C_3$ externally touching the circles $C$, $D$ & $C_2$ is calculated by setting the values of $a=1, b=1, c=r_2=\frac{1}{12}$ in the above standard formula as follows $$r_3=\frac{1\cdot1\cdot\frac{1}{12}}{2\sqrt{1\cdot1\cdot\frac{1}{12}\left(1+1+\frac{1}{12}\right)}+1\cdot1+1\cdot \frac{1}{12}+\frac{1}{12}\cdot 1}=\frac{1}{24}$$ Thus, continuing the same procedure, we can calculate radius of any circle $C_n$ using above standard formula.
