1.8

设 M(x)={M(M(x+11)),if x≤100,x−10,if x>100, M(x) = \left\{\begin{array}{ll} M(M(x+11)), & \text{if } x \leq 100, \\ x-10, & \text{if } x>100, \end{array} \right. 证明： M(x)={91,if x≤100,x−10,else. M(x) = \left\{\begin{array}{ll} 91, & \text{if } x \leq 100, \\ x-10, & \text{else.} \end{array} \right.

证明

只需证，当 0≤x≤1000\leq x \leq 100 时，M(x)=91M(x) = 91

  • M(90)=M(M(101))=M(91)=M(92)=⋯=M(100)=M(M(111))=M(101)=91M(90) = M(M(101))=M(91)=M(92) = \cdots = M(100) = M(M(111))=M(101)=91；
  • 当 0≤x≤1000\leq x \leq 100 时，存在 kk 使得 90≤x+11k≤10090 \leq x+11k \leq 100，从而 M(x)=M2(x+1∗11)=Mk+1(x+11k)=MkM(x+11k)=Mk(91)=91M(x) = M^2(x+1*11)= M^{k+1}(x+11k) = M^kM(x+11k)=M^k(91) = 91。