If a, b, c are in AP, b-a, c-b and a are in GP, then a: b: c is

Step-by-step solution

Given a, b, c are in AP, we have 2b = a + c . Let the common difference be d , so b = a + d and c = a + 2d . The terms b - a, c - b, a are in GP. Substituting the expressions in terms of a and d : b - a = (a + d) - a = d c - b = (a + 2d) - (a + d) = d The terms are d, d, a . Since d, d, a are in GP, the condition for GP is d^2 = d a . This implies d^2 - ad = 0 , so d(d - a) = 0 . Case 1: d = 0 . Then a = b = c , which gives the ratio 1: 1: 1 . This is not among the options. Case 2: d = a . Substituting d = a into t