You tell a person to try something. On the 1st & 2nd attempts, they succeed. On the 3rd attempt, they fail. They give up, saying "I failed, so I will always fail." From only 1 data point (of 3) they imagine a larger general trend that is false.
I am thinking it's Composition, with a larger set of actions being composed of individual actions, one of which is a failed attempt.
It's not always a logical fallacy. If you tell someone to lift a 2,000 lb weight, and they succeed at age 30 and age 50, and then fail at age 70, this person is perfectly justified in concluding that they will fail later. Induction is complicated, it involves general facts and implicit assumptions, and clarifying stupid trivial syllogisms does not help is learning how to do induction or logic. It is something that takes up a lot of time in philosophy however, and it does make people sound superficially smart.
It's not a logical fallacy. It's faulty inductive reasoning: as you rightly say, they infer a trend from a finite number of cases in an unwarranted manner.
For that matter, there are many cases in which three fails do not warrant the inference that the person will always fail (consider learning to ride a bicycle).
I really hate these kinds of questions when they're arbitrarily posed without a given list to work from. Is this for an assignment, or a general question? The problem here is that too many fallacies would fit the bill. Without a set of standing specific definitions to work from, what your person is doing by her statement is
...and so on.
If you just look at the statement alone, though, it's only making one mistake. I can demonstrate as a haiku:
I've x'ed in the past,
Thus I'll x in the future.
Gambler's Fallacy.
It's the classic Humean problem, that outcomes of the past dictate - or even relate - to outcomes in the future. The rest is just additional evidence, there to obfuscate the answer by distracting you from the logical statement. Suppressed evidence, and similar charges related to the additional information are only rhetorical fallacies, not necessarily logical ones.
I've come to the conclusion that it's a Hasty Generalization.
That fallacy involves going from an element or subset of a thing, in this case an attempt at doing something to deciding that a general trait exists e.g. failure or success.
This could be categorized as at least two different kinds of logical fallacy.
Composition. (you guessed it!). The phrase people generally use to describe the fallacious reasoning involved in the fallacy of composition is the untrue statement: "What is true of the part is true of the whole". So, if the "part" is the failed attempt in your example, the "whole" would be the set of all attempts. Just because the part entails failure, it does not follow that the sum of all parts entail failure.
Ad hominem - An ad hominem argument is one that is used to discredit a claimant's assertion based on some unrelated negative attribute of the claimant. This is like thinking because someone is wrong once, that they are therefore always wrong, and attacking a claim with an unknown truth value on that basis. This would apply to your example if the particular conclusion drawn is that the subject always fails at making an argument. E.g. "I made a math error, so my math will always be wrong"
