How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression? I have a progression that goes like this:
$$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$
Is there a way I can simplify it?

Yes.
\begin{align*}
1\cdot 2 + 2\cdot 3 + \dotsb + (n-1)\cdot n
&= 1\cdot(1+1) + 2\cdot(2+1) + \dotsb + (n-1)\cdot((n-1)+1) \\
&= 1^2 + 1 + 2^2 + 2 + \dotsb + (n-1)^2 + (n-1) \\
&= (1^2+2^2+\dotsb+(n-1)^2) + (1+2+\dotsb+(n-1))
\end{align*}
Both of those sums have well-known closed forms; see here, for example.
