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Continued or Mean Proportion

If a, b, c,...

Question

If $a, b, c, d$ are in continued proportion, prove that $a : b + d = c^{3} : c^{2} d + d^{3}$ .

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Solution

$\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k$ , then
$a = b k = c k^{2} = d k^{3} and b = c k = d k^{2} and c = d k$
L.H.S. $= \frac{a}{b + d} = \frac{d k^{3}}{d k^{2} + d} = \frac{k^{3}}{k^{2} + 1}$
R.H.S. $= \frac{c^{3}}{c^{2} d + d^{3}} = \frac{d^{3} k^{3}}{d^{2} k^{2} \cdot d + d^{3}} = \frac{k^{3}}{k^{2} + 1}$
$∴$ L.H.S. $=$ R.H.S.

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