Question

Pe mulțimea numerelor reale se definește legea de compoziție asociativă x*y = xy + x + y. Calculați 1 * 2 * 3 * 4 * ... * 2020.

Answer 1

$\begin{aligned}x*y &= xy+x+y\\ &=xy+x+y+1-1 \\ &=y(x+1)+(x+1)-1\\ &=(x+1)(y+1)-1\end{aligned}$

$1*2*3*4*...*2020 =\\ \\ =(1*2)*(3*4)*(5*6)*(7*8)*...*(2019*2020)\\ \\ = (2\cdot 3-1)*(4\cdot 5-1)*(6\cdot 7-1)*...*(2020\cdot 2021-1)\\ \\ =(2\cdot 3\cdot 4\cdot 5-1)*(6\cdot 7\cdot 8\cdot 9-1)*...*(2018\cdot 2019\cdot 2020\cdot 2021-1)\\ \\\vdots\,\,\text{(analog)}\\\\ =2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot ...\cdot 2018\cdot 2019\cdot 2020\cdot 2021-1\\ \\ = \boxed{2021!-1}$