\(A=\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{1}{4}\right)+\left(1-\dfrac{1}{8}\right)+.......................+\left(1-\dfrac{1}{1024}\right)\)
\(\Leftrightarrow A=\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{7}{8}+..................+\dfrac{1023}{1024}\)
\(\Leftrightarrow A=\dfrac{1}{2}+\dfrac{3}{2^2}+\dfrac{7}{2^3}+...............+\dfrac{1023}{2^{10}}\)
\(\Leftrightarrow2A=1+\dfrac{3}{2}+\dfrac{7}{2^2}+...............+\dfrac{1023}{2^9}\)
\(\Leftrightarrow2A-A=\left(1+\dfrac{3}{2}+\dfrac{7}{2^2}+............+\dfrac{1023}{2^9}\right)-\left(\dfrac{1}{2}+\dfrac{3}{2^2}+\dfrac{7}{2^3}+...............+\dfrac{1023}{2^{10}}\right)\)\(\Leftrightarrow A=1+\left(\dfrac{3}{2}-\dfrac{1}{2}\right)+\left(\dfrac{7}{2^2}-\dfrac{3}{2^2}\right)+.................+\left(\dfrac{1023}{2^9}-\dfrac{511}{2^9}\right)-\dfrac{1023}{2^{10}}\)\(\Leftrightarrow A=\left(1+1+...+1\right)-\dfrac{1023}{2^{10}}\)
\(\Leftrightarrow A=9-\dfrac{1023}{2^{10}}\)
