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

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

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

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

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

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

    QED
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