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limit sequence of polygons, trisecting sides

Source: Putnam 1984 B6

September 10, 2021
limitsgeometry

Problem Statement

A sequence of convex polygons (Pn),n≥0,(P_n),n\ge0, is defined inductively as follows. P0P_0 is an equilateral triangle with side length 11. Once PnP_n has been determined, its sides are trisected; the vertices of Pn+1P_{n+1} are the interior trisection points of the sides of PnP_n. Express lim⁡n→∞[Pn]\lim_{n\to\infty}[P_n] in the form ab\frac{\sqrt a}b, where a,ba,b are integers.
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