Abarim Publications' online Biblical Greek Dictionary
The particle αρα (ara) marks inference and means "then", "and so thus" or "hence". In the Greek classics this particle could also serve to mark a question (in English we change the word order for that, but Greek uses inflections, which leaves the order of words up to the author's good humor) but questions often served to mark an affirmative rather than an interrogative clause ("isn't such and such so? Well then!"). This suggests a proportional correlation between a negative question (is it not?) and a positive declaration (it is!), which explains the inferential use of our particle in the New Testament. It's used 35 times; see full concordance.
Obviously related to the above, the adverb or conjunction αρα (ara) marks both a question and an implied a negative answer: not!. This important negative nuance follows from the positive implication of a negative answer. The negative question, "Is not such and such true?" implies an affirmative: "Yes, it is true", which means that the positive question, "Is such and such true?" implies a negative, "No, it is not true".
This adverb of negative affirmation cannot be properly translated into English since English uses word order and question marks to mark questions — unlike, for instance, Slavic languages, which use li for that. And this has allowed very serious misunderstandings into common Christian dogma. When the disciples asked Jesus: "Who (ara) is the greatest in the Kingdom of Heaven?" (Matthew 18:1), the implied answer is "no one" in the sense of "no single one", since only the least are single individuals (Matthew 11:11) whereas the greater is the community (John 17:21, Acts 4:32, Ephesians 4:3-6). The question is not about how to determine rank but about how to create a perfect community: "Who then if no one is the greatest ..."
This adverb and the particle of inference are spelled identically, and their difference wasn't marked until the normalization of diacritics (breathing and pronunciation marks) in post-Biblical times. That means that it's not always perfectly clear whether we're dealing with our particle of inference, or our adverb of negative inquiry, and clarity must be derived from the context. But most scholars agree that our adverb is used 19 times; see full concordance.