Would you (mind/like to)...?, Why not...?
V-(으)시겠어요? is used when politely suggesting something to the listener or when asking about the listener's preference or intention.
It corresponds to 'Would you (mind/like to)...?' or 'Why not...?' in English and conveys a much more formal and polite feeling than -(으)ㄹ래요?/-(으)실래요?.
When the stem ends in a vowel, -시겠어요? is used, and when it ends in a consonant, -(으)시겠어요? is used.
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