Question

\(\frac{2}{15}\)+\(\frac{2}{35}\)+\(\frac{2}{63}\)+...+\(\frac{2}{99}\)+\(\frac{2}{399}\)

Answer 1

\(\frac{2}{15}+\frac{2}{35}+\frac{2}{63}+............+\frac{2}{399}\)

\(=\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+.......+\frac{2}{19.21}\)

\(=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+..........+\frac{1}{19}-\frac{1}{21}\)

\(=\frac{1}{3}-\frac{1}{21}=\frac{7}{21}-\frac{1}{21}=\frac{6}{21}=\frac{2}{7}\)

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Answer 2

\(\frac{2}{15}+\frac{2}{35}+...+\frac{2}{399}\)

\(=2\left(\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{19.21}\right)\)

\(=2\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{19}-\frac{1}{21}\right)\)

\(=2\left(\frac{1}{3}-\frac{1}{21}\right)\)

\(=2\times\frac{2}{7}=\frac{4}{7}\)

Answer 3

\(\frac{2}{15}+\frac{2}{35}+\frac{2}{63}+...+\frac{2}{399}\)

\(=\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{19.21}\)

\(=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{19}-\frac{1}{21}\)

\(=\frac{1}{3}-\frac{1}{21}\)

\(=\frac{2}{7}\)