Question

Tính B = 1.2.3 + 2.3.4 + ... + (n - 1)n(n + 1)

Answer 1

Áp dụng tính kế thừa ta có:

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

\(=1.2.3.4-0.1.2.3+2.3.4.5-1.2.3.4+...+\left(n-1\right)n\left(n+1\right)\left(n+2\right)\left[\left(n-2\right)\left(n-1\right)n\left(n+1\right)\right]\)\(=\left(n-1\right)n\left(n+1\right)\left(n+2\right)-0.1.2.3=\left(n-1\right)n\left(n+1\right)\left(n+2\right)\)

\(\Rightarrow B=\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}\)