about gerund: 哪一句子是正確的？

Both sentences (1) and (3) are correct grammatically.
in (1) checking is like a noun and it is modified by an adjective- easy
in (3) checking is like a verb and it is modified by an adverb - easily.
Undoubtedly (1) is a common expression, used more frequently than (3)In linguistics a gerundis a kind of verbal noun.It behaves as a verb withina clause and thereforea gerund may be modified by an adverb or/and havean object.e.g.She stopped crying bitterly.The verbalcharacteristics of the gerund include the following:1. The gerund of transitive verbs can take a direct object: I've made good progress in speaking English.. 2. The gerund can be modified by an adverb: Breathing deeply helps you to calm down. 3.The gerund has the distinctions of aspect and voice. Having read the book once before makes me more prepared. Being deceived can make someone feel angry. It is an interesting question whether gerunds are morelike nouns or verbs. And the answer is that they can be treated as nouns, andthey can be treated as verbs.
- Verb-like gerund: I like running the church bazaar well (here runninghas an object like a verb, and is qualified by an adverb-“well”).
- Noun-like gerund: I like his excellent running of the church bazaar (hererunning does not have an object, and is qualified by an adjective, likea noun).

(3) is correct.

Please present your membership card for checking easily.

Easily = Adv.
Checking = N. / Gerund



Hope that you will find it useful.

only (1)
cannot be (2),(3)