Suppose that John ate $x$ peanuts on the fourth night. Since he ate 6 more peanuts each night than on the previous night, then he ate $x-6$ peanuts on the third night, $(x-6)-6=x-12$ peanuts on the second night, and $(x-12)-6=x-18$ peanuts on the first night. Since John ate 120 peanuts in total, then $x+(x-6)+(x-12)+(x-18)=120$, and so $4x-36=120$ or $4x=156$ or $x=39$. Therefore, John ate 39 peanuts on the fourth night.