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Determine if 0, point, 2, 9, 2, 9, 9, 2, 9, 9, 9, 2, 9, 9, 9, 9, 2, 9, 9, 9, 9, 9, 2, dot dot dot0.292992999299992999992... is rational or irrational and give a reason for your answer.
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GPT-5 mini
The decimal is
x = 0.292992999299992999992...,
which is the concatenation of blocks B_k = (2 followed by k nines) for k = 1,2,3,...
Thus for every n there is a block B_n containing n consecutive 9's, so x has arbitrarily long runs of consecutive 9's.
If x were rational its decimal expansion would be eventually periodic: beyond some place N the digits would repeat with some fixed period p, so beyond N the lengths of runs of identical digits would be bounded by p. That contradicts the fact that x has runs of 9's of every finite length. (The usual ambiguity of a terminating vs. repeating-9 expansion does not apply here, because the digits are not eventually all 9's — there are infinitely many 2's.)
Therefore x is irrational.
x = 0.292992999299992999992...,
which is the concatenation of blocks B_k = (2 followed by k nines) for k = 1,2,3,...
Thus for every n there is a block B_n containing n consecutive 9's, so x has arbitrarily long runs of consecutive 9's.
If x were rational its decimal expansion would be eventually periodic: beyond some place N the digits would repeat with some fixed period p, so beyond N the lengths of runs of identical digits would be bounded by p. That contradicts the fact that x has runs of 9's of every finite length. (The usual ambiguity of a terminating vs. repeating-9 expansion does not apply here, because the digits are not eventually all 9's — there are infinitely many 2's.)
Therefore x is irrational.
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