The progression has 1st term 256 =   and the common ratio r = .


Hence, the n-th term is   =  =  = .


We look for the term   < 0.001,  or   < 0.001.


Take logarithm base 2 of both sides


    9-n < 

    9 - n < -9.96...


Since we work with integer numbers, we need to solve

    9 - n <= -10


It is equivalent

    n >= 9 + 10

    n >= 19.


ANSWER.  First term of the GP, which is less than 0.001, is the 19-th term   =  = 0.000977...   ANSWER