Final answer:
The most likely mode of inheritance for the male-killing trait in beetles that can be cured with antibiotics and involves an extra 16s rRNA is infectious inheritance.
Step-by-step explanation:
The mode of inheritance most likely at play in a beetle species where a male-killing trait favors female offspring is infectious inheritance. This conclusion is supported by two observations: firstly, strains that exhibit the male-killing trait can be treated using antibiotics, which indicates the involvement of a microbial agent. Secondly, the presence of an extra 16s rRNA detected by PCR in affected females suggests the presence of an additional prokaryotic organism, likely a bacterium, which is not normally part of the beetle's cellular structure. In contrast to modes such as maternal inheritance and organellar inheritance, which imply genetic transmission via ova, infectious inheritance would involve the horizontal transfer of infection, potentially from mother to offspring or within the community. Sex-limited inheritance is not consistent with the cure by antibiotics, hence that and the other options are less likely than infectious inheritance.