Question 23
Given a finite sequence $S = (a_1, a_2, \ldots, a_n)$ of $n$ real numbers, let $A(S)$ be the sequence $$
\small \left(\frac{a_1 + a_2}{2}, \frac{a_2 + a_3}{2}, \ldots, \frac{a_{n-1} + a_n}{2}\right)
$$ 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 $x \gt 0,$ and let $S = (1, x, x^2, \ldots, x^{100}).$ If $A^{100}(S) = (\frac{1}{2^{50}}),$ then what is $x?$
\small \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 $x \gt 0,$ and let $S = (1, x, x^2, \ldots, x^{100}).$ If $A^{100}(S) = (\frac{1}{2^{50}}),$ then what is $x?$