• M0★ M1★ M2★ M3★ M4★ M5★ M6

    The hint is a bit confusing because the question only has four 1s, and the hint says the number is equal to 0.25+0.00111111...+0.0000001111. From the question, I thought the number was 0.251111222222222222222222222222222222222222222.... \(\text{\tiny{you get the idea}}\), but I think the hint is saying that the number is 0.2511111111111111111111111....

  • ADMIN M0★ M1 M5

    @divinedolphin You were right to initially think that the decimal was \(0.251111\overline{2222}\ldots.\) 🙂 The multiple \(2\)'s come from the overlap of the first set of \(1\)'s in \(0.\textcolor{blue}{00}\overline{\textcolor{red}{1111}111111}\ldots\) plus the second set of \(1\)'s in \(0.\textcolor{blue}{00}\textcolor{red}{0000}\overline{1111}\ldots \)

    The first set of \(1\)'s is shifted over to the right by two spaces, allowing the \(25\) to appear.
    The second set of \(1\)'s is shifted to the right by \(6\) spaces, allowing the four \(\textcolor{red}{\text{red } 1}\)'s from the \(0.\textcolor{blue}{00}\overline{\textcolor{red}{1111}111111}\ldots\) to show through.
    And the remaining \(2\)'s come from the places that have \(1\)'s in both the \(0.\textcolor{blue}{00}\overline{\textcolor{red}{1111}111111}\ldots\) and the \(0.\textcolor{blue}{00}\textcolor{red}{0000}\overline{1111}\ldots \)