Question

Rút gọn: (1/2*5)+(1/5*8)+(1/8*11)+...+(1/(3n+2)*(3n+5))

Answer 1

Đặt \(A=\dfrac{1}{2\cdot5}+\dfrac{1}{5\cdot8}+\dfrac{1}{8\cdot11}+...+\dfrac{1}{\left(3n+2\right)\left(3n+5\right)}\)

\(3A=\dfrac{3}{2\cdot5}+\dfrac{3}{5\cdot8}+\dfrac{3}{8\cdot11}+...+\dfrac{3}{\left(3n+2\right)\left(3n+5\right)}\)

\(3A=\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{11}+...+\dfrac{1}{3n+2}-\dfrac{1}{3n+5}\)

\(3A=\dfrac{1}{2}-\dfrac{1}{3n+5}\)

\(3A=\dfrac{3n+3}{2\left(3n+5\right)}\)

\(A=\dfrac{n+1}{6n+10}\)