Let the positive numbers a₁, a₂, a₃, ...., a₃ₙ be in GP. If P is the GM of a₁, a₂, a₃, ...., aₙ and Q is the GM of aₙ₊₁, aₙ₊₂, aₙ₊₃, ...., a₃ₙ, then what is the GM of 3n numbers?

GM of all 3n terms = (product of all 3n terms)^(1/3n). Product of all = (product of first n)(product of last 2n). Pⁿ = product of first n, so product of first n = Pⁿ. Q^(2n) = product of last 2n, so product of last 2n = Q^(2n). Total product = Pⁿ · Q^(2n). GM = (Pⁿ Q^(2n))^(1/3n) = P^(1/3) Q^(2/3).