\(A=1\cdot2+2\cdot3+3\cdot4+...+n\left(n+1\right)\)

\(3A=1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+...+3n\left(n+1\right)\)

\(=1\cdot2\cdot\left(3-0\right)+2\cdot3\cdot\left(4-1\right)+3\cdot4\cdot\left(5-2\right)+...+n\left(n+1\right)\left[\left(n+2\right)-\left(n-1\right)\right]\)\(=1\cdot2\cdot3-0\cdot1\cdot2+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+...+n\left(n+1\right)\left(n+2\right)-\left(n-1\right)n\left(n+1\right)\)\(=n\left(n+1\right)\left(n+2\right)\)

\(\Rightarrow A=\dfrac{n\left(n+1\right)\left(n+2\right)}{3}\)