Question

 

Báo cáoTính B = 1.2.3 + 2.3.4 + ... + (n - 1)n(n + 1)

Answer 1

S = 1.2.3 + 2.3.4 +..+ (n-1).n.(n+1) 

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

(k-1)k(k+1).4 = (k-1)k(k+1)[(k+2) - (k-2)] = (k-1)k(k+1)(k+2) - (k-2)(k-1)k(k+1) 

1.2.3.4 = 1.2.3.4 
2.3.4.4 = 2.3.4.5 - 1.2.3.4 
3.4.5.4 = 3.4.5.6 - 2.3.4.5 

(n-2)(n-1)n.4 = (n-2)(n-1)n(n+1) - (n-3)(n-2)(n-1)n 
(n-1)n(n+1).4 = (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1) 

4S = (n-1)n(n+1)(n+2) 

=> S = (n-1)n(n+1)(n+2)/4 

Answer 2

4(1.2.3) = 1.2.3.4 - 0.1.2.3
4(2.3.4) = 2.3.4.5 - 1.2.3.4
4(3.4.5) = 3.4.5.6 - 2.3.4.5
....................................
4(n-1)n(n+1) = (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1)
=> 4 B = (n-1)n(n+1)(n+2) => B= (n-1)n(n+1)(n+2):4

Answer 3

\(4B=4.\left[1.2.3+2.3.4+3.4.5+...+\left(n-1\right).n.\left(n+1\right)\right]\)

\(4B=1.2.3.4+2.3.4.4+3.4.5.4+...+\left(n-1\right).n.\left(n+1\right).4\)

\(4B=1.2.3.4+2.3.4.\left(5-1\right)+3.4.5.\left(6-2\right)+...+\left(n-1\right).n.\left(n+1\right).\left[\left(n+2\right)-\left(n-2\right)\right]\)

\(4B=1.2.3.4+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+...+\left(n-1\right).n.\left(n+1\right).\left(n-2\right)-\)

\(\left(n-2\right).\left(n-1\right).n.\left(n+1\right)\)

\(4B=\left(n-1\right).n.\left(n+1\right).\left(n+2\right)\)

\(B=\left[\left(n-1\right).n.\left(n+1\right).\left(n+2\right)\right]:4\)