Question

tinh

1.2+2.3+3.4+........+N(N+1)

Answer 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

= 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + n(n + 1)[(n + 2) - (n - 1)]

= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + .... + n(n + 1)(n + 2) - (n - 1)n(n + 1)

= n(n + 1)(n + 2)

\(\Rightarrow A=\frac{N\left(N+1\right)\left(N+2\right)}{3}\)