Question

S=1*2+2*3+3*4+...+n*(n+1).

Answer 1

$S = 1\cdot 2+2\cdot 3+3\cdot 4+4\cdot 5 +...+(n-1)\cdot n+n\cdot (n+1)\\ S = 2\cdot(1+3)+4\cdot(3+5)+...+n(n-1+n+1)\\ S = 2\cdot 4+4\cdot 8+...+n\cdot 2n \\ S = 2\cdot (2\cdot 2+4\cdot 4+6\cdot 6+...n\cdot n)\\ S = 2\cdot (2^2+4^2+6^2+...+n^2) \\ S = 2\cdot 2^2 \cdot \Big(1^2+2^2+3^2+...+(\frac{n}{2})^2\Big)\\ S = 2^3\cdot \dfrac{\frac{n}{2}(\frac{n}{2}+1)(2\cdot\frac{n}{2}+1)}{6} \\ S = 8\cdot \dfrac{\frac{1}{2\cdot 2\cdot 2}\cdot n(n+2)(2n+2)}{6} \\ S = 8\cdot \dfrac{2n(n+2)(n+1)}{8\cdot 6}$

$S = \dfrac{n(n+1)(n+2)}{3}$