3. Variant 1. Find the largest root of the equation

$(x+1)(x+2)-(x+2)(x+3)+(x+3)(x+4)-(x+4)(x+5)+\cdots-(x+1000)(x+1001)=0$.
Answer: -501.

Solution. Let's break the terms into pairs of adjacent terms and factor out the common factor, we get $(x+2)(x+1-x-3)+\cdots+(x+1000)(x+999-x-1001)=-2 \cdot(x+2+x+4+\cdots+x+1000)=$ $-2 \cdot\left(500 x+\frac{2+1000}{2} \cdot 500\right)=-1000 \cdot(x+501)=0$. From this, $x=-501-$ the only root.