If log e a , log e b , log e c are in an A . P . and log e a - log e 2 b , log e 2 b - log e 3 c , log e 3 c - log e a are also in an A . P . , then a : b : c is equal to

Step-by-step solution

Given: log a , log b , log c are in A . P . ⇒ 2 log b = log a + log c ⇒ b 2 = a c . . . 1 And log a - log 2 b , log 2 b - log 3 c , log 3 c - log a are in A . P ⇒ 2 log 2 b 3 c = log a 2 b + log 3 c a ⇒ 2 b 3 c 2 = a 2 b × 3 c a ⇒ 2 b 3 c 2 = 3 c 2 b ⇒ 2 b = 3 c ⇒ 4 b 2 = 9 c 2 & 6 b = 9 c . . . 2 Now solving the equation 1 & 2 we get, 4 a c = 9 c 2 ⇒ 4 a = 9 c . . . 3 Then from equation 2 & 3 we get, 4 a = 6 b = 9 c = k ⇒ a = k 4 , b = k 6 & c = k 9 ⇒ a : b : c = 1 4 : 1 6 : 1 9 = 9 : 6 : 4