Let the positive numbers a_{1},a_{2},a_{3},.....,a_{3n} be in GP. If P is the GM of a_{1},a_{2},a_{3},....,a_{n} and Q is the GM of a_{n+1},a_{n+2},a_{n+3},....,a_{3n}, then what is the GM of 3n numbers?

Explanation

P represents the GM of n terms, so its contribution to the product of all terms is P^n. Q represents the GM of 2n terms, so its contribution is Q^{2n}. The GM of all 3n terms is the 3n-th root of their total product: (P^n Q^{2n})^{\frac{1}{3n}} = P^{1/3}Q^{2/3}.

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