As a result or consequence of something. He didn't wake up early. Consequently, he was late to work. The word 'consequently' is correct and usable in written English. You can use it to introduce a fact that has been caused by what has just been mentioned in a sentence or a preceding sentence.

At the hotel, we take safety very seriously, and consequently, we have security officers on duty twenty-four hours a day. Robert did not like the brownie; consequently, he threw it in the trash. Consequently, Jim is not passing his classes because he is not completing his assignments.