@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 \(0.\textcolor{blue}{00}\textcolor{red}{0000}\overline{1111}\ldots \)

🙂

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