Find the Sum $1\cdot2+2\cdot3+\cdots + (n-1)\cdot n$ Find the sum $$1\cdot2 + 2\cdot3 + \cdot \cdot \cdot + (n-1)\cdot n.$$
This is related to the binomial theorem.  My guess is we use the combination formula . . .
$C(n, k) = n!/k!\cdot(n-k)!$
so . . . for the first term $2 = C(2,1) = 2/1!(2-1)! = 2$
but I can't figure out the second term $3 \cdot 2 = 6$ . . . 
$C(3,2) = 3$ and $C(3,1) = 3$ I can't get it to be 6.
Right now i have something like . . .
$$ C(2,1) + C(3,2) + \cdot \cdot \cdot + C(n, n-1) $$
The 2nd term doesn't seem to equal 6.
What should I do? 

Check for a relationship between $n (n + 1) (n + 2)$ and your sum by checking small cases and prove it by induction.
