I went to the hospital and I saw a patient being treated last night.
What kind of sentence it is being treated in the above sentence.
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The phrase "being treated" includes a form of the verb to be followed by something that's traditionally called a past participle, which you're calling v3. This is a passive voice construction, similar to constructions like "was treated", "is treated" and "will be treated".
The form of the verb to be that's used here is the -ing form. Traditionally, -ing forms are labeled as either gerunds or present participles, depending on their use. Some modern grammars call this the gerund-participle form, since the form is the same regardless of its use.
Different frameworks use different labels to describe the same thing. Depending on framework, the phrase "being treated" in this sentence might be labeled as a passive present participial phrase, a passive continuous participial phrase, or even (I suspect) a passive non-finite clause.
No matter what label you choose, you're looking at a non-finite verb construction that has no tense, but that has the passive voice and the continuous aspect.