How to solve this series ?

Question

How to Solve series  like this

n(n-1)+ (n-1)(n-2) + (n-2)(n-3) + ...........

Answer 1

$n(n-1)+(n-1)(n-2)+(n-2)(n-3)+...$

$n(n-1) +(n-1)((n-1)-1)+(n-2)((n-2)-1)+.....$

$=n^2-n + (n-1)^2- (n-1) + (n-2)^2 - (n-2)+...$

$=(n^2 +(n-1)^2 + (n-2)^2 +... ) -  (n +(n-1) + (n-2) +... )$

$=\frac{n(n+1)(2n+1)}{6} - \frac{n(n+1)}{2}$

$=\frac{n(n^2-1)}{3}$

Note:

$\sum ^n_{i=1} i= \frac{n(n+1)}{2}$

$\sum ^n_{i=1} i^2= \frac{n(n+1)(2n+1)}{6}$

Comments

Thank you very much!