Ta có : A = 1 . 2 + 2 . 3 + 3 . 4 + … + n(n + 1)

3A = 1 . 2 . (3 – 0) + 2 . 3 . (4 – 1) + 3 . 4 . (5 – 2) + ... + n(n + 1)[(n + 2) – (n – 1)]

3A = 1.2.3 – 0.1.2 + 2.3.4 – 1.2.3 + 3.4.5 – 2.3.4 + .... + n(n + 1)(n + 2) – (n – 1)n(n + 1)

3A = (1.2.3 –1.2.3) + (2.3.4 – 2.3.4) + ... + [(n – 1)n(n + 1) – (n – 1) n(n + 1)] + n(n + 1)(n + 2)

3A = n(n + 1)(n + 2)

$\text{A}=\frac{n\left(n+1\right)\left(n+2\right)}{3}$.