\(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^9}\)
=>\(2A=1+\frac{1}{2}+...+\frac{1}{2^8}\)
=>\(2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^8}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^9}\right)\)
\(=1-\frac{1}{2^9}=\frac{511}{512}\)
\(B=\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+\frac{1}{108}+\frac{1}{324}+\frac{1}{972}\)
=>\(3B=\frac{3}{4}+\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+\frac{1}{108}+\frac{1}{324}\)
=>\(3B-B=\left(\frac{3}{4}+\frac{1}{4}+\frac{1}{12}+...+\frac{1}{324}\right)-\left(\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+...+\frac{1}{972}\right)\)
=>\(2B=\frac{3}{4}-\frac{1}{972}=\frac{182}{243}\)
=>\(B=\frac{182}{243}:2=\frac{91}{243}\)
