HMMT 二月 2001 · 几何 · 第 3 题

HMMT February 2001 — Geometry — Problem 3

Square ABCD is drawn. Isosceles Triangle CDE is drawn with E a right angle. Square DEF G is drawn. Isosceles triangle F GH is drawn with H a right angle. This process is repeated infinitely so that no two figures overlap each other. If square ABCD has area 1, compute the area of the entire figure.

解析

Square ABCD is drawn. Isosceles Triangle CDE is drawn with E a right angle. Square DEF G is drawn. Isosceles triangle F GH is drawn with H a right angle. This process is repeated infinitely so that no two figures overlap each other. If square ABCD has area 1, compute the area of the entire figure. Solution: Let the area of the n th square drawn be S and the area of the n th triangle n √ be T . Since the hypotenuse of the n th triangle is of length S , its legs are of length n n √ 2 S S l S 1 2 n n n l = , so S = l = and T = = . Using the recursion relations, S = and n +1 n n n − 1 2 2 2 4 2 ( ) 1 1 1 1 1 5 1 T = , so S + T = + = + 2 = . Thus the total area of the figure is n n +1 n n n − 1 n +1 n n 2 2 2 2 2 2 2 ∞ ∞ ∑ ∑ 5 1 5 S + T = = . n n n 2 2 2 n =1 n =1