Question

2/3+2/15+2/35+2/63+.....+2/143

Answer 1

\(\frac{2}{3}+\frac{2}{15}+\frac{2}{35}+\frac{2}{63}+...+\frac{2}{143}\)

\(=\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+\frac{2}{7\times9}+...+\frac{2}{11\times13}\)

\(=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{11}-\frac{1}{13}\)

\(=1-\frac{1}{13}=\frac{12}{13}\)

Answer 2

Đặt \(A=\frac{2}{3}+\frac{2}{15}+\frac{2}{35}+...+\frac{2}{143}\)

\(\Rightarrow A=\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+...+\frac{2}{11.13}\)

\(\Rightarrow A=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{11}-\frac{1}{13}\)

\(\Rightarrow A=1-\frac{1}{13}=\frac{13}{13}-\frac{1}{13}=\frac{2}{13}\)