\(B=\frac{2}{15}+\left(\frac{5}{9}+\frac{-6}{9}\right)\)

\(\Leftrightarrow B=\frac{2}{15}+\frac{-1}{9}\)

\(\Leftrightarrow B=\frac{12}{90}+\frac{-10}{90}\)

\(\Leftrightarrow B=\frac{2}{90}\)

\(\Leftrightarrow B=\frac{1}{45}\)

\(B=\frac{2}{15}=\left(\frac{5}{9}+\frac{-6}{9}\right)\)

\(B=\frac{2}{15}+\frac{-1}{9}\)

\(B=\frac{18}{135}+\frac{-15}{135}\)

\(B=\frac{3}{135}\)

\(B=\frac{2}{15}+\left(\frac{5}{9}+-\frac{6}{9}\right)\)

\(B=\frac{2}{15}+-\frac{1}{9}\)

\(B=\frac{1}{45}\)

\(Linh\)\(Yukiko\)

\(T=\frac{4}{2.4}+\frac{4}{4.6}+\frac{4}{6.8}+...+\frac{4}{2008.2010}\)

\(T=2.\left(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+...+\frac{2}{2008.2010}\right)\)

\(T=2.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2008}-\frac{1}{2010}\right)\)

\(T=2.\left(\frac{1}{2}-\frac{1}{2010}\right)\)

\(T=2.\frac{502}{1005}=\frac{1004}{1005}\)

\(\Rightarrow T=\frac{1004}{1005}\)

\(A=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2007.2009}+\frac{1}{2009+2011}\)

\(A=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{2009+2011}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2009}-\frac{1}{2011}\right)\)

\(A=\frac{1}{2}.\left(1-\frac{1}{2011}\right)\)

\(A=\frac{1}{2}.\frac{2010}{2011}\)

\(\Rightarrow A=\frac{1005}{2011}\)

\(C=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{100}\right)\)

\(C=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{99}{100}\)

\(C=\frac{1.2.3...99}{2.3.4...100}\)

\(\Rightarrow C=\frac{1}{100}\)

\(A=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{10}}\)

\(3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^9}\)

\(3A-A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^9}-\frac{1}{3}-\frac{1}{3^2}-...-\frac{1}{3^{10}}\)

\(2A=1-\frac{1}{3^{10}}\)

\(A=\frac{1-\frac{1}{3^{10}}}{2}\)