Given a finite sequence S=\left(a_{1}, a_{2}, \ldots, a_{n}\right) of n real numbers, let A(S) be the sequence
\left(\dfrac{a_{1}+a_{2}}{2}, \dfrac{a_{2}+a_{3}}{2}, \ldots, \dfrac{a_{n-1}+a_{n}}{2}\right)
of n-1 real numbers. Define A^{1}(S)=A(S) and, for each integer m, 2 \leq m \leq n- 1 , define A^{m}(S)=A\left(A^{m-1}(S)\right). Suppose x>0, and let S=\left(1, x, x^{2}, \ldots, x^{100}\right). If A^{100}(S)=\left(1 / 2^{50}\right), then what is x ?
Answer Choices
A. 1-\dfrac{\sqrt{2}}{2}
B. \sqrt{2}-1
C. \dfrac{1}{2}
D. 2-\sqrt{2}
E. \dfrac{\sqrt{2}}{2}