Suppose a piano tuner hears 2 beats per second when listening to the combined sound from her tuning fork and the piano note being tuned. After slightly tightening the string, she hears 1 beat per second. Should she loosen or should she further tighten the string?

further tightening is required.

Explanation:

The beat created / sec = difference of frequencies

Initial beat heard = 2

so difference of frequencies = 2

after tightening beat heard = 1

difference of frequencies decreases because frequency of tuning fork was higher than piano sound.

on further tightening difference decreases because tightening increases the frequency of piano hence further tightening is required for resonance.