Đặt
\(A=1\cdot2\cdot3+2\cdot3\cdot4+3\cdot4\cdot5+4\cdot5\cdot6+.......+n\left(n+1\right)\left(n+2\right)\)\(4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+3\cdot4\cdot5\cdot4+.......+n\left(n+1\right)\left(n+2\right)\cdot4\)\(4A=1\cdot2\cdot3\cdot\left(4-0\right)+2\cdot3\cdot4\cdot\left(5-1\right)+3\cdot4\cdot5\cdot\left(6-2\right)+........+n\left(n+1\right)\left(n+2\right)\left(n+3-n-1\right)\)\(4A=1\cdot2\cdot3\cdot4-0+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+....+n\left(n+1\right)\left(n+2\right)\left(n+3\right)-\left(n-1\right)n\left(n+1\right)\left(n+2\right)\)\(4A=n\left(n+1\right)\left(n+2\right)\left(n+3\right)\)
\(A=\dfrac{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}{4}\)
Vậy \(A=\dfrac{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}{4}\)
