\(f(x) = \left\lfloor\frac{2 - 3x}{x + 3 + 2x}\right\rfloor\)

f(1) = floor(-1/6) = -1

f(2) = floor(-4/9) = -1

f(3) = floor(-7/12) = -1

f(4) = floor(-10/15) = -1

...

f(10) = floor(-28/33) = -1

...

f(100) = floor(-298/303) = -1

... f(x) = floor((-3x + 2) / (3x + 3)), which the numerator will never be able to be -3x + 3, if it did it would be -1, so the floor'd always be -1.

Hence, f(1) + f(2) + ... f(1000) = -1 * 1000 = **-1000**