Learn Turkish: Sen mektup yazıyorsun. - You are writing a letter.

This is the personal subject suffix for sen (you, singular). In the present continuous tense, it attaches directly to the -yor suffix. Because the vowel 'o' in -yor never changes, the 'you' suffix that follows it will always be -sun to match it.

Why is there an ı in the middle of yazıyorsun?

The verb root is yaz- (to write), which ends in a consonant. The present continuous suffix is -yor. To connect them, we need a vowel. Based on the 4-way vowel harmony rules, the 'a' in yaz- means the connecting vowel must be ı.

The English translation says 'a letter', but there is no word for 'a' (bir) in the Turkish sentence. Why is that?

In Turkish, if you are doing a general action like 'letter-writing', the object (mektup) just sits bare right before the verb. You do not need bir unless you want to emphasize that you are writing exactly one specific letter.

Since the verb ending already means 'you', do I have to use the word Sen?

No, you can just say Mektup yazıyorsun. Turkish often drops subject pronouns like sen or ben because the verb endings already tell us who is doing the action. Using Sen here just adds a little emphasis.