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

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Solution

Given $a, b, c, d$ are in continued proportion
$⟹ \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k ($ say $)$
$⟹ c = d k, b = c k = k^{2} d, a = b k = k^{3} d$
$R H S = (\frac{a - b}{b - c})^{3} = (\frac{k^{3} d - k^{2} d}{k^{2} d - k d})^{3} = k^{3}$
$L H S = \frac{a}{d} = k^{3}$
$L H S = R H S$
Hence Proved

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