This is not a real situation; please ignore legal issues that you might think apply, because they don't.
Let's say I have a set of 200 known valid license keys for a hypothetical piece of software's licensing algorithm, and a license key consists of 5 sets of 5 alphanumeric case-insensitive (all uppercase) characters. Example: HXDY6-R3DD7-Y8FRT-UNPVT-JSKON
Is it possible (or likely) to extrapolate other possible keys for the system?
What if the set was known to be consecutive; how do the methods change for this situation, and what kind of advantage does this give?
I have heard of "keygens" before, but I believe they are probably made by decompiling the licensing software rather than examining known valid keys. In this case, I am only given the set of keys and I must determine the algorithm. I'm also told it is an industry standard algorithm, so it's probably not something basic, though the chance is always there I suppose.
If you think this doesn't belong in Stack Overflow, please at least suggest an alternate place for me to look or ask the question. I honestly don't know where to begin with a problem like this. I don't even know the terminology for this kind of problem.
In general, the answer is, "No, you can't do anything useful."
If the people generating the keys got lazy and failed to use some sort of cryptographic-quality hash off of an index number (with sufficient bit-mixing to thwart any inspection on your part), then you might assume some sort of functional form of random number generation and see if you can back out, for example, a modulus for a linear congruential random number generator, or a series of bit-mixing shifts and adds and such as in the Jenkins hash function or whatever.
There are no algorithms for going from some generic structure that you spot to the algorithm that produces said structure; something akin to this is what you appear to be asking for. (Such algorithms are provably impossible in general; if you want the simplest algorithm that can compute your keys, the problem is isomorphic to the computation of Kolmogorov complexity, which is fiendishly hard ("effectively impossible thus far") to compute.)