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sequence

Source: Netherlands 1992

June 28, 2009
limitalgebra proposedalgebra

Problem Statement

We consider regular n n-gons with a fixed circumference 4 4. Let rn r_n and an a_n respectively be the distances from the center of such an n n-gon to a vertex and to an edge. (a) (a) Determine a4,r4,a8,r8 a_4,r_4,a_8,r_8. (b) (b) Give an appropriate interpretation for a2 a_2 and r2 r_2 (c) (c) Prove that a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n) and r_{2n}\equal{}\sqrt{a_2n r_n}. (d) (d) Define u_0\equal{}0, u_1\equal{}1 and u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1}) for n n even or u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}} for n n odd. Determine lim⁡n→∞un \displaystyle\lim_{n\to\infty}u_n.
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