\(A=2\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+......+\frac{1}{87}-\frac{1}{90}\right)\)
\(\Rightarrow A=2\left(\frac{1}{15}-\frac{1}{90}\right)\)
\(\Rightarrow A=2.\frac{1}{18}=\frac{1}{9}\)
\(A=\frac{6}{15.18}+\frac{6}{18.21}+\frac{6}{21.24}+...+\frac{6}{87.90}\)
\(=6.\frac{1}{3}.\left(\frac{3}{15.18}+\frac{3}{18.21}+\frac{3}{21.24}+...+\frac{3}{87.90}\right)\)
\(=2.\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+\frac{1}{21}-\frac{1}{24}+...+\frac{1}{87}-\frac{1}{90}\right)\)
\(=2.\left(\frac{1}{15}-\frac{1}{90}\right)\)
\(=2.\frac{1}{18}\)
\(=\frac{1}{9}\)
\(B=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{120}\)
\(=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{240}\)
\(=\frac{2}{4.5}+\frac{2}{5.6}+\frac{2}{5.7}+...+\frac{2}{15.16}\)
\(=2.\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{15.16}\right)\)
\(=2.\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{15}-\frac{1}{16}\right)\)
\(=2.\left(\frac{1}{4}-\frac{1}{16}\right)\)
\(=2.\frac{3}{16}\)
\(=\frac{3}{8}\)
a) \(A=\frac{6}{15.18}+\frac{6}{18.21}+...+\frac{6}{87.90}\)
\(A=\frac{3.6}{3.15.18}+\frac{3.6}{3.18.21}+...+\frac{3.6}{3.87.90}\)
\(A=2\left(\frac{3}{15.18}+\frac{3}{18.21}+...+\frac{3}{87.90}\right)\)
\(A=2\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+...+\frac{1}{87}-\frac{1}{90}\right)\)
\(A=2\left(\frac{1}{15}-\frac{1}{90}\right)\)
\(A=2.\frac{1}{18}=\frac{1}{9}\)
b)
\(B=2\left(\frac{1}{20}+\frac{1}{30}+.....+\frac{1}{240}\right)\)
\(\Rightarrow B=2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+.....+\frac{1}{15}-\frac{1}{16}\right)\)
\(\Rightarrow B=2\left(\frac{1}{4}-\frac{1}{16}\right)\)
\(\Rightarrow B=2.\frac{3}{16}=\frac{3}{8}\)