Help to compute sum of products I need to compute the following sum:
$$(1\times2\times3)+(2\times3\times4)+(3\times4\times5)+ ...+(20\times21\times22)$$
All that I have deduced is:


*

*Each term is divisible by $6$. So sum is is divisible by $6$.

*Sum is divisible by $5$ as 1st term is $1$ less than multiple of $5$ and second term is $1$ more than multiple of $5$. Next three terms are divisible by $5$. This cycle continues for every $5$ terms.


So sum will obviously be divisible by $30$.

HINT:
$$(r-1)r(r+1)=r^3-r$$
Now, $$\sum_{r=1}^n r=\frac{n(n+1)}2$$ and  $$\sum_{r=1}^n r^3=\left(\frac{n(n+1)}2\right)^2$$
Here $r=1$ to $21$
