Let a_ 1 , a_ 2 , , a_ n be positive real numbers in a G.P. for each n , let A_ n , G_ n…

Mathematics · JEE Main · NTA Exams — Progression and Series

Let \(a_{1}, a_{2}, \ldots, a_{n}\)be positive real numbers in a G.P. for each \(\text { n }\), let \(A_{n}, G_{n}, H_{n}\) be respectively, the arithmetic mean, geometric mean and harmonic mean of \(a_{1}, a_{2}, \ldots, a_{n}\). Find the expression for the geometric mean of \(G_{1}, G_{2}, G_{3}, \ldots, G_{n}\) in terms of \(A_{1}, A_{2}, \ldots, A_{n}\) and \(H_{1}, H_{2}, \ldots, H_{n}\).

\(\left[\mathbf{A}_{1} \cdot \mathbf{A}_{2} \cdot \ldots \cdot \mathbf{A}_{n} \cdot H_{1} \cdot H_{2} \cdot \ldots \cdot H_{n}\right]^{\frac{n}{2}}\)
\(\left[A_{1} \cdot A_{2} \cdot \ldots \cdot A_{n} \cdot H_{1} \cdot H_{2} \cdot \ldots \cdot H_{n}\right]^{\frac{1}{2 n}}\)
\(\left[\mathbf{A}_{1} \cdot \mathbf{A}_{2} \cdot \ldots \cdot \mathbf{A}_{n} \cdot H_{1} \cdot H_{2} \cdot \ldots \cdot H_{n}\right]^{\frac{1}{n}}\)
None of these

Answer

(C) [ A _ 1 A _ 2 A _ n H_ 1 H_ 2 H_ n ]^ 1 n

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Let a_ 1 , a_ 2 , , a_ n be positive real numbers in a G.P. for each n , let A_ n , G_ n…

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