\(Ax2=1+\dfrac{1}{2}+\dfrac{1}{4}+....+\dfrac{1}{512}\)
\(Ax2-A=\left(1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{512}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{1024}\right)\)
\(A=1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{512}-\dfrac{1}{2}-\dfrac{1}{4}-\dfrac{1}{1024}\)
\(A=1-\dfrac{1}{1024}=\dfrac{1023}{1024}\)