i will do it next week还是 i will do it in the next week? next

i will do it next week正确
在this, that, next, last, every, yesterday, tomorrow等构成的时间状语前不加任何介词.
（last year, this month, next week, last Saturday, every day, yesterday afternoon, tomorrow morning）
[误]We're going to study harder in this term.
[正] We're going to study harder this term.
[误] I'll be free on next Sunday.?
[正] I'll be free next Sunday

next week 用在现在将来时的句子中
the next week 用在过去将来时的句子中
所以这句话第一句是对的

这里next week 直接做状语，时 间状语。eg.I am home !就不可以说成I am at home .因为这里也是home 直接做了状语！