Question

tinh

1.2+2.3+3.4+.....+n(n+1)

Answer 1

1.2 + 2.3 + 3.4 + ..... + n(n + 1)

\(=\frac{1.2.3+2.3.3+3.4.3+...+n\left(n+1\right).3}{3}\)

\(=\frac{1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+n\left(n+1\right)\left[\left(n+2\right)-\left(n-1\right)\right]}{3}\)

\(=\frac{1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+..+n\left(n+1\right)\left(n+2\right)-\left(n-1\right)n\left(n+1\right)}{3}\)

\(=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)