MathDB

Given a finite sequence S \equal{} (a_1,a_2,\ldots,a_n) of

n

real numbers, let

A(S)

be the sequence \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right) of n \minus{} 1 real numbers. Define A^1(S) \equal{} A(S) and, for each integer

m

, 2\le m\le n \minus{} 1, define A^m(S) \equal{} A(A^{m \minus{} 1}(S)). Suppose

x > 0

, and let S \equal{} (1,x,x^2,\ldots,x^{100}). If A^{100}(S) \equal{} (1/2^{50}), then what is

x

? (A) 1 \minus{} \frac {\sqrt {2}}2\qquad (B) \sqrt {2} \minus{} 1\qquad (C) \frac 12\qquad (D) 2 \minus{} \sqrt {2}\qquad (E) \frac {\sqrt {2}}2

Problem Statement