\(\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+...+\dfrac{1}{998\cdot999\cdot1000}\)

Apply formula : \(\dfrac{2n}{a\left(a+n\right)\left(a+2n\right)}=\dfrac{1}{a\left(a+n\right)}-\dfrac{1}{\left(a+n\right)\left(a+2n\right)}\)

\(2.B=\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+...+\dfrac{2}{998\cdot999\cdot1000}\)

\(2.B=\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{998\cdot999}-\dfrac{1}{999\cdot1000}\)

\(2.B=\dfrac{1}{1\cdot2}-\dfrac{1}{999\cdot1000}\)

\(2B=\dfrac{499499}{999000}\)

\(\Rightarrow B=\dfrac{499499}{1998000}\)