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  • one year ago

Which positive real number \(x\) has the property that \(x\), \(\lfloor x \rfloor\), and \(x - \lfloor x\rfloor\) form a geometric progression (in that order)? (\(\lfloor x\rfloor\) means the greatest integer less than or equal to \(x\).)

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  1. anonymous
    • one year ago
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    Assume that \( \lfloor{x}\rfloor=1\). Then the common ratios become\[\frac{ 1 }{ x } = x-1\]Multiplying through by \(x\) and rearranging gives\[x^2-x-1=0\]Solving this quadratic yields two solutions, only one of which satisfies the question.

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