Should one write:
Anyone who studies regularly will do better and enjoy it more.
Anyone who studies regularly will do better and enjoys it more.
To make the sentence grammatically correct?
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To take the discussion in the comments to an answer...
"Anyone who studies regularly will do better and enjoys it more." is wrong, because the placement of "will do" becomes what "enjoy" must match, not "studies."
E.g., -Anyone who studies regularly will do better. -Anyone who studies regularly will enjoy it more.
Changing up the sequence, in English, really can change up the meaning, because (most of the time) verbs want to agree with the closest verb preceding them.
So you have to have: "Anyone who studies regularly will do better and enjoy it more."
If you use FumbleFingers' example, "Anyone who studies regularly enjoys it more and will do better," notice that sentence is matching "studies/enjoys," and the "will do" is connected by a conjunction that allows it to break tense from the prior verbs. (However, I would avoid using this sentence because "studies regularly enjoys" could be interpreted two ways: "anyone who studies, regularly enjoys it more" vs. "anyone who studies regularly, enjoys it more." Making your reader parse out the intended meaning will slow the reader down, when you want to carry the reader smoothly to your desired conclusion.)