Ratio of terms in continued proportion – UGC NET Question

For a, b and c to be in continued proportion, we have a:b = b:c, which implies b² = ac. Given a:b = 2:5, we can take a = 2k and b = 5k. Then b² = ac gives (5k)² = 2k·c, so 25k² = 2kc and c = (25k²)/(2k) = 25k/2. Therefore, c:b = (25k/2):(5k) = 25:10 = 5:2.

Option A:
Option A is correct because it results from applying the definition of continued proportion and simplifying c:b. It shows that c must be larger than b when a:b is 2:5, which aligns with the idea that b is a geometric mean between a and c and hence lies between them.

Option B:
Option B, 2:5, simply restates the ratio a:b and would imply that c is smaller than b, contradicting the fact that c should lie on the opposite side of b from a in the geometric mean setting. Using 2:5 for c:b does not satisfy b² = ac with the given a:b.

Option C:
Option C, 4:5, comes from a possible misreading that c is closer to b than a is, but if c:b were 4:5, the product ac would no longer equal b². Substituting 4:5 fails the algebraic condition linking the three terms in continued proportion.

Option D:
Option D, 5:4, suggests that c is only slightly larger than b, but the actual computation shows c is 25k/2 while b is 5k, making c significantly larger. The ratio 5:4 therefore understates c and does not satisfy the required equality b² = ac.