Exactly, it's completely arbitrary. I like to frame it in terms of fitting n points (the existing data in the problem) to a polynomial of degree n+1, where there's an infinite number of ways to pick a solution and still satisfy the initial data.

Maybe the "solution with the lowest Kolmogorov complexity".

In a sibling comment, I replied that usually a repeating pattern can also be applied, but that one usually requires storing the n-sequence, rarely making it the shortest encodable rule.

I think it's better phrased as "find the best rule", with a tacit understanding that people mostly agree on what makes a rule decent vs. terrible (maybe not on what makes one great) and a tacit promise that the sequence presented has at least one decent rule and does not have multiple.

A rule being "good" is largely about simplicity, which is also essentially the trick that deep learning uses to escape no-free-lunch theorems.

This is exactly what I hated about these kind of questions!!

Or just, the pattern simply repeats for the n next number.. like come on, there is no objective metric for which rule is "better".