'book' or 'books'?
I suspect that it should be 'book', because the full sentence should be 'the first book and the third book'.
Right?
It'll be books in this case.
While that's decent reasoning, it's not quite right. The full sentence would be
the first book and the third book
Two separate books and you're considering both of them in
the first and the third books
....so, you use the plural.
In your example, you would generally use the plural
The first and the third books are...
The first and third books are...