Which of the following words relates to apathy and a lack of energy?

The term that relates to apathy and a lack of energy is "lethargy." This word embodies a state of sluggishness, both physically and mentally, where an individual may feel a pronounced absence of motivation or interest in engaging with activities. Lethargy often implies a level of fatigue or drowsiness that affects a person's ability to feel excited or enthusiastic about tasks they normally would find engaging. This connection makes "lethargy" the appropriate choice to describe a state characterized by a lack of energy and apathy, as it specifically conveys those qualities.

The other words present a contrasting notion. "Excitement," "enthusiasm," and "vivacity" all imply a sense of energy or liveliness, suggesting engagement rather than apathy. They describe states or traits that are characterized by active participation and emotional involvement, which is opposite to the feeling of lethargy.