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For which values of n will 2n be a multiple of ten?
I conjecture that there will be no value of n that will give an answer that is a multiple of ten. This is because, each number in the sequence is double the answer before. To be able to have a multiple of ten, the last digit in the number before would have to be a five. This is impossible as a five would never come up in the sequence as no whole number x 2 = 5.
For example, if you took a random gues that 1000 would be somewhere in the sequence, keep halfing the previous numbers eg. 1000/2=500/2=250/2=125(note the 5)/2=62.5... Eventually the 5 turns up in the sequence, in which case it is I correct, the number before that is not a whole number.
It could be, that you could double a 0 to get a multiple of ten. This would work, but how could you get to the first zero? From the previous proof, it explains that it is not possible to get to the 5 digit number before the zero,because you cannot double numbers that aren't whole and 5 is a prima factor and aren't in the base 2 sequence. Therefore, all the numbers before it are unreachable.