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In the geometric sequence 256,128,64,32,…. which is the first term that is less than 0.001?
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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
