Hãy rút gọn phép tính: 1 × 2 + 2 × 3 + 3 × 4 +…+ n × (n + 1).

Đặt A = 1 × 2 + 2 × 3 + 3 × 4 +…+ n × (n + 1)

3A = 1 × 2 × 3 + 2 × 3 × 3 + 3 × 4 × 3 +…+ n × (n + 1) × 3

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 × (n + 1) × (n – 1)

3A = n × (n + 1) × (n + 2)

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