Question

a) Chứng minh k(k+1)(k+2)-k(k-1)(k+1)=3k

b) Áp dụng để tính: S=1.2+2.3+3.4+...+n(n+1)

Answer 1

a: \(k\left(k+1\right)\left(k+2\right)-k\left(k-1\right)\left(k+1\right)\)

\(=k\left(k+1\right)\left(k+2-k+1\right)\)

=3k(k+1)

b: \(S=1\cdot2+2\cdot3+...+n\left(n+1\right)\)

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

\(=\left(1+2+...+n\right)+\left(1^2+2^2+...+n^2\right)\)

\(=\dfrac{n\left(n+1\right)}{2}+\dfrac{n\left(n+1\right)\left(2n+1\right)}{6}\)

\(=\dfrac{3n\left(n+1\right)+n\left(n+1\right)\left(2n+1\right)}{6}=\dfrac{n\left(n+1\right)\left(2n+4\right)}{6}=\dfrac{n\left(n+1\right)\left(n+2\right)}{3}\)