Geometric Progression

A geometric progression has nth term ar^{n-1}, where a is the first term and r the common ratio, a structure JEE Advanced exploits through ratios of terms. Dividing the sixth term by the third term eliminates a: (ar^5)/(ar^2) = r^3 = 96/12 = 8. Taking the real cube root gives r = 2, which is positive and therefore consistent with the requirement of positive terms. Option 3 fails because 3^3 = 27, not 8. Option 4 gives 4^3 = 64, also inconsistent with the ratio 8. Option 1/2 is the reciprocal and would force decreasing terms, contradicting the term values that increase from 12 to 96. Hence r = 2. Plausibility check: with r = 2 and third term 12, the first term is 12/4 = 3, and the sixth term is 3 × 2^5 = 3 × 32 = 96, exactly matching the given data and confirming the geometric ratio.