Question

[nguyễn hoàng lê thi]

C/m với n€ N^* a) 1.4 + 4.7 + 7.10 +....+ (3n -2)(3n +1) = n(n +1)^2

Answer 1

[Trần Minh Hoàng]

Đặt \(T=1.4+4.7+...+\left(3n-2\right)\left(3n+1\right)\).

Ta có: \(9T=1.4.\left[7-\left[-2\right]\right]+4.7.\left(10-1\right)+7.10.\left(13-4\right)+...+\left(3n-2\right).\left(3n+1\right).\left[\left(3n+4\right)-\left(3n-5\right)\right]=1.4.7-\left(-2\right).1.4+4.7.10-1.4.7+7.10.13-4.7.10+...+\left(3n-2\right)\left(3n+1\right)\left(3n+4\right)-\left(3n-5\right)\left(3n-2\right)\left(3n+1\right)=\left(3n-2\right)\left(3n+1\right)\left(3n+4\right)-\left(-2\right).1.4=9n\left(3n^2+3n-2\right)\Rightarrow T=n\left(3n^2+3n-2\right)\).