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Let \( a_{1}, a_{2}, \ldots \ldots . ., a_{n} \) be positive real numbers in geometric progression. For each \( n \), let \( A_{n}, G_{n}, H_{n} \) be respectively the arithmetic mean, geometric mean \& harmonic mean of \( a_{1}, a_{2}, \ldots \ldots ., a_{n} \). Prove that
\( \mathrm{P} \) \( G=\prod_{k=1}^{n}\left(A_{k} \quad H_{k}\right)^{\frac{1}{2 n}} \), Where \( G \) is geometric mean between \( G_{1}, G_{2}, \ldots \ldots . ., G_{n} \).

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