Answer
690 Hz.
Work Step by Step
The two sources are in phase, since the sound is loudest at points that are the same distance from each of the two sources. For destructive interference, the difference in distance from the sources, 0.25 meters, equals an odd number of half wavelengths (page 342).
$$0.25m=\lambda/2$$
$$\lambda=0.50m$$
$$f=\frac{v}{\lambda}=\frac{343m/s}{0.50m}\approx 690Hz$$
If we had assumed that the difference in distance from the sources, 0.25 meters, equaled 3, 5, 7, etc. half wavelengths, then the frequencies would be higher than the specified range of 500-1000 Hz.
