\(A=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+..........+\frac{1}{1024}\)
\(2A=2.\left(1+\frac{1}{2}+\frac{1}{4}+.........+\frac{1}{1024}\right)\)
\(2A=2+1+\frac{1}{2}+...........+\frac{1}{512}\)
\(2A-A=\left(2+1+\frac{1}{2}+...........+\frac{1}{512}\right)-\left(1+\frac{1}{2}+\frac{1}{4}+..............+\frac{1}{1024}\right)\)
\(2A-A=2-\frac{1}{1024}\)
\(A=\frac{2048}{1024}-\frac{1}{1024}\)
\(A=\frac{2047}{1024}\)