If on january 1,2015 you decide to read one thousand and one night over the next 1,001 night an which date would you finally finish

Starting January 1, 2015, there are 364 remaining nights.
From 1,001 nights, it will be 637 nights left.

In 2016, there are 366 nights. Subtract it to 637 nights left, you will get 271 nights left.
In 2017, 271 is lesser than 365.
From January to August, the nights left is 28.

So the date will be September 28, 2017.