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Proportion

If q is the m...

Question

If q is the mean proportional between p and r prove that :

$\frac{p^{3} + q^{3} + r^{3}}{p^{2} q^{2} r^{2}} = \frac{1}{p^{3}} + \frac{1}{q^{3}} + \frac{1}{r^{3}}$

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Solution

Since q is the mean proportional between p and r
$q squared equals p r L. H. S equals fraction numerator p cubed plus q cubed plus r cubed over denominator p squared q squared r squared end fraction equals fraction numerator p cubed plus left parenthesis p r right parenthesis q plus r cubed over denominator p squared left parenthesis p r right parenthesis r squared end fraction equals fraction numerator p cubed plus p q r plus r cubed over denominator p cubed r cubed end fraction equals 1 over r cubed plus fraction numerator q over denominator p squared r squared end fraction plus 1 over p cubed equals 1 over r cubed plus q over open parentheses q squared close parentheses squared plus 1 over p cubed equals 1 over r cubed plus 1 over q cubed plus 1 over p cubed R. H. S$

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p^{3} {(q - r)}^{3} + q^{3} {(r - p)}^{3} + r^{3} {(p - q)}^{3}

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