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Increasing Sequence or Dncreasing Sequence

Source: 1985 National High School Mathematics League, Exam One, Problem 6

February 23, 2020

Problem Statement

Let 0<a<10<a<1. x1=a,x2=ax1,⋯ ,xn=axn−1x_1=a,x_2=a^{x_1},\cdots,x_n=a^{x_{n-1}}. Then sequence (xn)(x_n) (A)\text{(A)} Is an increasing sequence. (B)\text{(B)} Is an decreasing sequence. (C)\text{(C)} Increases when nn is odd, decreases when nn is even. (D)\text{(D)} Decreases when nn is odd, increases when nn is even.
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