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Question
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
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Solution
a, b and c are in G.P.
\[\therefore b^2 = ac\] .......(1)
\[\text { LHS } = a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right)\]
\[ = \frac{b^2 c^2}{a} + \frac{a^2 c^2}{b} + \frac{a^2 b^2}{c}\]
\[ = \frac{\left( ac \right) c^2}{a} + \frac{\left( b^2 \right)^2}{b} + \frac{a^2 \left( ac \right)}{c} \left[ \text { Using } (1) \right]\]
\[ = a^3 + b^3 + c^3 = \text { RHS }\]
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