If you end a sentence with an abbreviation, do you place an additional period after it?

When an abbreviation with a period ends a sentence, that period will suffice to end the sentence. For example:

Jane Doe lives in Washington, D.C.

When a question ends with an abbreviation, end the abbreviation with a period and then add the question mark.

Didn't he use to live in Washington, D.C.?

The Period