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Math Help - Find max value - (Triangles, geometric progressions, greatest integer(floor)function)

  1. #1
    Super Member fardeen_gen's Avatar
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    Find max value - (Triangles, geometric progressions, greatest integer(floor)function)

    If for r > 1, three successive terms of a Geometric Progression with common ratio r represent sides of a triangle, then find the maximum value of (\lfloor 2r \rfloor + \lfloor - r \rfloor)
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  2. #2
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    Triangle Inequality

    Take some triangle with side lengths a,ar,ar^2 for some a>0 and r>1. Without loss of generality, let a=1 and find the allowable values of r for a triangle to exist:

    (i) 1+r>r^2 true for all r>1 but r<\phi for \phi the golden ratio \approx 1.618
    (ii) 1+r^2>r true for all r>1
    (iii) r+r^2>1 true for all r>1

    Therefore, r\in(1,\phi) to satisfy this condition. It can be shown that:

    \lfloor 2r \rfloor + \lfloor -r \rfloor = <br />
\left\{ \begin{array}{rcl}<br />
0 & \mbox{if} & 1<r<1.5<br />
\\1 & \mbox{if} & 1.5\leq r<2<br />
\\2 & \mbox{if} & 2\leq r<2\phi<br />
\end{array}\right.

    Therefore the max value of this function on the allowable interval is 2.

    QED
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