썼겠느냐 << 쓰(다) + 었 + 겠(다) + 느냐
쓰다 : to write, to use...
았/었 + 겠다 : grammar pattern
느냐 : 해라체 interrogative ending suffix(가다->가느냐, 먹다->먹느냐)
Adjective/Verb + 았/었 + 겠다 : This grammar pattern is used to convey supposition, conjecture, a guess, a thought or observation about something in the past. It can sometimes be translated as 'I think', 'I bet', 'I guess', 'It must have been ~'. This pattern can not be used to express the thoughts or supposition (etc) of a second or third person and must express the thought of the person saying or writing it.
무거웠겠다. It must have been heavy for you.
애인이랑 좋은시간 보냈겠어요? You must have had good time wity your boyfriend, right?
걔들이 살아있었으면 22살이 되었겠구나. They must have been 22, if they were not dead.
지금쯤 모두 잠이 들었겠다. Everyone must have been asleep.
몰라서 안 썼겠느냐?
You/He/She/Someone must have not written it despite knowing that, right?