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Ratio and Proportion — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsRatio and Proportion4 Marks
Question
If $a, b, c$ are in continued proportion, prove that:
$
\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{a}{b^2 c^2}+\frac{b}{c^2 a^2}+\frac{c}{a^2 b^2}
$
✓

Answer

As $a, b, c$, are in continued proportion
$
\begin{aligned}
& \text { Let } \frac{a}{b}=\frac{b}{c}= k \\
& \text { L.H.S. }=\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3} \\
& =\frac{1}{\left(c k^2\right)^3}+\frac{1}{(c k)^3}+\frac{1}{c^3} \\
& =\frac{1}{c^3 k^6}+\frac{1}{c^3 k^3}+\frac{1}{c^3} \\
& =\frac{1}{c^3}\left[\frac{1}{k^6}+\frac{1}{k^3}+\frac{1}{1}\right] \\
& \text { R.H.S. }=\frac{a}{b^2 c^2}+\frac{b}{c^2 a^2}+\frac{c}{a^2 b^2} \\
& =c \frac{k^2}{(c k)^2 c^2}+\frac{ ck }{c^2\left(c k^2\right)^2}+\frac{c}{\left(c k^2\right)^2(c k)^2} \\
& =\frac{c k^2}{c^4 k^2}+\frac{ ck }{c^4 k^4}+\frac{c}{c^4 k^6} \\
& =\frac{1}{c^3}+\frac{1}{c^3 k^3}+\frac{1}{c^3 k^6}
\end{aligned}
$
$\begin{aligned} & =\frac{1}{c^3}\left[1+\frac{1}{k^3}+\frac{1}{k^6}\right] \\ & =\frac{1}{c^3}\left[\frac{1}{k^6}+\frac{1}{k^3}+1\right] \\ & \therefore \text { L.H.S. = R.H.S. }\end{aligned}$

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