Question
If $\frac{a}{c}=\frac{c}{d}=\frac{c}{f}$ prove that $: \frac{\left(a^3+c^3\right)^2}{\left(b^3+d^3\right)^2}=\frac{e^6}{f^6}$
✓
Answer
$\begin{aligned} & \frac{a}{c}=\frac{c}{d}=\frac{c}{f}= k \text { (say) } \\ & \therefore a = bk , c = dk _{ r } e = fk \\ & \text { L.H.S. } \frac{\left(a^3+c^3\right)^2}{\left(b^3+d^3\right)^2} \\ & =\frac{\left(b^3 k^3+d^3 k^3\right)^2}{\left(b^3+d^3\right)^2} \\ & =\frac{\left[k^3\left(b^3+d^3\right)\right]^2}{\left(b^3+a^3\right)^2} \\ & =\frac{k^6\left(b^3+d^3\right)^2}{\left(b^3+d^3\right)^2}= k ^6 \\ & \text { R.H.S. }=\frac{e^6}{f^6} \\ & =f^6 \frac{k^6}{f^6}= k ^6 \\ & \therefore \text { L.H.S. }=\text { R.H.S. }\end{aligned}$
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