1. In 1924, she finally entered flight school.
2. In 1924, she finally entered the flight school.
3. In 1924, she finally entered a flight school.
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Which one is grammatical or commonly used?
All are correct, but they don't all mean exactly the same thing.
1 and 3 mean any flight school, in general.
2 means a particular flight school that both the speaker and the listener know of.