Let f(x) = \left\lfloor\frac{3x - 5}{3x + 4}\right\rfloor. Evaluate f(1)+f(2) + f(3) + \dots + f(999)+f(1000). (This sum has $1000$ terms, one for the result when we input each integer from $1$ to $1000$ into $f$.)
All the other terms evaluate to 0 because the inside function will approach (but never reach) y = 1 since this is the horizontal asymptote of that function
+866 
