Which of these sentences is correct? and why?
There is a car and a bicycle in the yard.
There are a car and a bicycle in the yard.
Which of these should be used in an academic essay?
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The correct usage here is "are", as "there" is not the subject; rather, this is known as "expletive construction", and the subject of the sentence is "a car and a bicycle". Check out my link for a full explanation.
In common speech, however, American native speakers will frequently say "is" for exactly the reason you mention in your comment above; we translate it internally as "There is a car and there is a bicycle in the yard". In fact, I can't shake the feeling that using "are" sounds weird when I hear it, no matter how I rationalize it.
Hope this helps.