Proof of a bizarre identity

Question

Prove That

$$ \prod_{r=1}^{n} (1-x^r) = 1 - \sum_{r=1}^{n} (1-x)\cdot(1-x^2)\cdot \ldots\cdot(1-x^{r-1})x^r $$

I found this identity through experimentation with Wolfram Alpha while trying to find a formula for the product on the LHS.

I have been able to prove it using induction, but I'm interested in how one would derive such identities. I'm seeking a direct proof without induction.

Any help will be appreciated.
Thanks.

Answer 1

Hint. One may just see it as a telescopic sum.

By observing that $$ \prod_{r=1}^{k} (1-x^r)-\prod_{r=1}^{k+1} (1-x^r)=\left(\prod_{r=1}^{k} (1-x^r)\right)x^{k+1}. $$