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Math Help - Brain Teaser #2

  1. #1
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    Brain Teaser #2

    5.101001000100001....


    he decimal number aboce consists of only one and 0 to the right of the decimal point. The first is followed by one 0, the second 1 is followed by two 0, the third 1 is followed by three 0 and so on. What is the total number of 0s between the 98th and the 101st 1 in this decimal number?


    what i supposed was that 98th 1 has 98 0, and so on, thus i added up 98+99+100=297.... but for some reason the asnwer is 291. I need a good explanation, and some tips to solve number pattern problems and sequence problems. Thank you very much for your help!
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  2. #2
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    Hello,

    The n-th 1 is preceded by a number of 0 equal to S_n=1+2+\dots+(n-1)=\sum_{k=1}^{n-1} k=\frac{n(n-1)}{2}

    The p-th 1 is preceded by a number of 0 equal to S_p=1+2+\dots+(p-1)=\sum_{k=1}^{p-1} k=\frac{p(p-1)}{2}

    Hence, assuming that p>n, there is a number of 0 between the n-th and the p-th equal to :

    S_p-S_n=\frac 12 (p(p-1)-n(n-1))

    This is the theorical part..

    Here, p=101 and n=98... And I find the same result
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