If $a, b, c, d$ are in continued proportion, prove that $a : d =$ tripicated ratio of $(a - b) : (b - c)$ .

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Solution

$a : d =$ Duplicate rate of $(a - b) (b - c)$

$L H S = (\frac{a}{d})^{3} = s q r t \frac{a}{d} \frac{(d h^{3})^{3}}{d^{3}} = \sqrt{\frac{d h^{3}}{d}} \frac{d^{3} h^{9}}{d^{3}} = \sqrt{h^{3}} = h R H S \frac{a - b}{b - c} = \frac{d h^{3} - d h^{2}}{d h^{2} - d h} = \frac{d h^{2} (h - 1)}{d h (h - 1)} = h$

$∴ L H S = R H S$

Proved

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