Question

Find the following sum and prove your claim: (1*2)+(2*3)+..+n(n+1).

Answer 1

1*2+2*3+3*4+...+n(n+1)=
(1²+1)+(2²+2)+(3²+3)+...+(n²+n)=
(1² + 2² + 3² + ... + n²) + (1+2+3+...+n)=
$\frac{n(n+1)(2n+1)}{6} * \frac{n(n+1)}{2}$ =
$\frac{n(n+1)(n+2)}{3}$