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?

You are considering $$S=\sum_{i=1}^{n-1}i\,(i+1)=\sum_{i=1}^{n-1}i^2+\sum_{i=1}^{n-1}i$$ which are quite standard summations. For the first one $$\sum_{i=1}^{n-1}i^2=\frac{1}{6} (n-1) n (2 n-1)$$ and for the second $$\sum_{i=1}^{n-1}i=\frac{1}{2} (n-1) n$$ Add them together and simplify (noticing some nice factorizations).
