NL-NWO 1992 Problem 5

We consider regular $n$ -gons with a fixed circumference $4$ . Let $r_n$ and $a_n$ respectively be the distances from the center of such an $n$ -gon to a vertex and to an edge. $(a)$ Determine $a_4,r_4,a_8,r_8$ . $(b)$ Give an appropriate interpretation for $a_2$ and $r_2$ $(c)$ Prove that $a_{2n}=\frac{1}{2} (a_n+r_n)$ and $r_{2n}=\sqrt{a_2n r_n}.$ $(d)$ Define $u_0=0, u_1=1$ and $u_n=\frac{1}{2}(u_{n-2}+u_{n-1})$ for $n$ even or $u_n=\sqrt{u_{n-2} u_{n-1}}$ for $n$ odd. Determine $\lim_{n\to\infty}u_n$ .