Question

1x2x3+2x3x4+3x4x5+...+(n-1)xnx(n+1)

 

 

 

 

 

 

Answer 1

Đặt A = 1.2.3 + 2.3.4 + 3.4.5 +...+ (n - 1)n(n + 1)

4A = 1.2.3.4 + 2.3.4.4 + 3.4.5.4 +...+ (n - 1)n(n + 1).4

4A = 1.2.3.(4 - 0) + 2.3.4.(5 - 1) + 3.4.5.(6 - 2) +....+ (n - 1)n(n + 1).[(n + 2) - (n - 2)]

4A = 1.2.3.4 - 0.1.2.3 + 2.3.4.5 - 1.2.3.4 + 3.4.5.6 - 2.3.4.5 +...+ (n - 1)n(n + 1)(n + 2) - (n - 2)(n - 1)n(n + 1)

4A = [1.2.3.4 + 2.3.4.5 + 3.4.5.6 +....+ (n - 1)n(n + 1)(n + 2)] - [0.1.2.3 + 1.2.3.4 + 2.3.4.5 + (n - 2)(n - 1)n(n + 1)]

4A = (n - 1)n(n + 1)(n + 2) - 0.1.2.3

4A = (n - 1)n(n + 1)(n + 2)

=> A = \(\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}\)