Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$
of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\ldots ,x^{100})$. If $A^{100}(S)=(1/2^{50})$, then what is $x$?
给定由 $n$ 个实数组成的有限序列 $S=(a_1,a_2,\ldots ,a_n)$，令 $A(S)$ 为由 $n-1$ 个实数组成的序列
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$。
定义 $A^1(S)=A(S)$，并对每个整数 $m$（$2\le m\le n-1$）定义 $A^m(S)=A(A^{m-1}(S))$。设 $x>0$，令 $S=(1,x,x^2,\ldots ,x^{100})$。若 $A^{100}(S)=(1/2^{50})$，求 $x$。