\(A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{97.99}\)
\(A=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\)
\(A=1-\dfrac{1}{99}\)
\(A=\dfrac{98}{99}\)
\(B=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...+\dfrac{2}{8.9.10}\)
\(B=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{8.9}-\dfrac{1}{9.10}\)
\(B=\dfrac{1}{1.2}-\dfrac{1}{9.10}\)
\(B=\dfrac{1}{2}-\dfrac{1}{90}\)
\(B=\dfrac{22}{45}\)
\(A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+.........+\dfrac{2}{97.99}\)
\(\Leftrightarrow A=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+........+\dfrac{1}{97}-\dfrac{1}{99}\)
\(\Leftrightarrow A=1-\dfrac{1}{99}=\dfrac{98}{99}\)
\(B=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+.......+\dfrac{2}{8.9.10}\)
\(\Leftrightarrow B=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+......+\dfrac{1}{8.9}-\dfrac{1}{9.10}\)
\(\Leftrightarrow B=\dfrac{1}{1.2}-\dfrac{1}{9.10}\)
\(\Leftrightarrow B=\dfrac{1}{2}-\dfrac{1}{90}\)
\(\Leftrightarrow B=\dfrac{22}{45}\)
