Geometric Progression Properties

Representing three numbers in geometric progression symmetrically as a/r, a, ar simplifies product conditions, a standard JEE Advanced parametrisation. Their product is (a/r)(a)(ar) = a^3 = 216, so a = 6, which is the middle term directly. The remaining condition a/r + ar = 30 gives 6(1/r + r) = 30, hence r + 1/r = 5, leading to r^2 - 5r + 1 = 0 with valid positive roots, confirming consistency. Option 8 would force a^3 = 512, contradicting the product 216. Option 9 gives a^3 = 729, also inconsistent. Option 12 misreads the product as 1728. Hence the middle number is 6. The symmetric labelling a/r, a, ar is powerful because the unknown ratio r cancels in the product, so the middle term is pinned down even before the second condition is used, a separation of conditions JEE Advanced rewards. Plausibility check: the geometric mean of the outer two numbers must equal the middle term, and since their product is a^2 = 36, the geometric mean is 6, independently verifying the central value regardless of the particular ratio r chosen.