HMMT 二月 2001 · 几何 · 第 3 题

英文解析

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. Thisprocess is repeated infinitely so that no two figures overlap each other. If square ABCD hasarea 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 trianglen
   √
   be T . Since the hypotenuse of the n th triangle is of length S , its legs are of lengthn n
   √
   S S l S 12
   n n n 2
   l = , so S = l = and T = = . Using the recursion relations, S = andn +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 isn n +1 n n n − 1 n +1 n n
   2 2 2 2 2 2 2
   ∞ ∞
   ∑ ∑
   5 1 5
   S + T = = .
   n nn
   2 2 2
   n =1 n =1