Step-by-step explanation:
In the sentence "The student gathered signatures for increasing the library," the phrase "increasing the library" is a (C) Gerund phrase.
Gerund phrases are formed by a gerund (a verb form ending in "-ing" that functions as a noun) plus any modifiers or complements. In this case, "increasing" is a gerund that acts as the main verb of the gerund phrase, and "the library" is the direct object of the gerund. The entire phrase "increasing the library" functions as the object of the preposition "for" and describes the purpose or goal for which the student gathered signatures.