Learn Farsi: مگر نمی‌آیی؟ چرا، می‌آیم!

You are right that چرا usually means "why". However, when someone asks you a negative question (like "Aren't you coming?"), you use چرا instead of بله to say "yes, I am" to contradict the negative assumption. It is a very common and natural way to reply!

What exactly does مگر add to the first sentence?

You could just ask نمی‌آیی؟, but adding مگر at the start shows surprise or doubt. It is used when you expect one thing but see another, translating roughly to "Don't tell me you aren't coming!" or emphasizing the "Aren't you...?" feeling.

How does the verb نمی‌آیی break down?

It is the negative present continuous form of آمدن (to come). It is made of the negative prefix نـ, the present continuous prefix میـ, the verb stem آ, and the "you" (singular) ending یی.

Why are there two ی letters at the end of نمی‌آیی?

The present stem for this verb is آ (which is a vowel). The standard ending for "you" (singular) is ی. To prevent the vowel sounds from clashing awkwardly, Farsi inserts an extra ی to act as a bridge or glide sound between them.