I have a sentence :

Having finished the term paper, he begun studying for the exam.

I think "Having finished the term paper" is a reduced adjective clause. My question is what the original sentence according to TOEFL grammar is ?

I have no idea what TOEFL grammar says about this but I would say:

After he had finished the term paper, he began studying for the exam.

Hi Welcome.

In form, it's a present perfect participle clause, like: Having completely consumed all the groceries in the house, John had to go to the supermarket.
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