A few minutes ago I started answering one of those "how does this sequence continue" question to which the standard answer is, essentially, "any way you like". The question was deleted while I was composing this answer. Perhaps there should be some generic such answer we can point to each time this kind of question recurs.
Edit in response to comments and answers.
The answer below is indeed too snarky, standing alone. In any particular case I would first try to guess the intended context and address that, then append the discussion pointing out that without more information there really are infinitely many answers.
There are many questions in math classes and on this site and in math riddles that present a finite sequence of integers and ask you determine how the sequence continues.
The only correct mathematical answer is that any continuation is possible.
Often the asker (perhaps a teacher) does not know that there is no single correct answer. There may be some context that helps you figure out what the asker had in mind.
One natural first step is to look at the differences, and then the differences of the differences. When these become constant there is a polynomial that generates the finite sequence you start with. Perhaps that's the continuation the poser had in mind.
When I was growing up in New York the classic riddle was to extend the sequence $$ 8,14,23,34, \ldots $$ The answer is $42$ - the next stop on the Manhattan branch of the Brighton Beach subway line.