Question

If \( \log_e a, \log_e b, \log_e c \) are in an A.P. and \( \log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a \) are also in an A.P., then \( a : b : c \) is equal to

• A. 9 : 6 : 4
• B. 16 : 4 : 1
• C. 25 : 10 : 4
• D. 6 : 3 : 2

Answer 1

The Correct Option is A

Since \( \log_e a, \log_e b, \log_e c \) are in an A.P., we have:  
\(b^2 = ac\)

Also, since \( \log_e \left( \frac{a}{2b} \right), \log_e \left( \frac{2b}{3c} \right), \log_e \left( \frac{3c}{a} \right) \) are in an A.P., we get:  
\(\left( \frac{2b}{3c} \right)^2 = \frac{a}{2b} \times \frac{3c}{a}\)

\(\implies \frac{b}{c} = \frac{3}{2}\)
Substituting into equation (1):  
\(b^2 = a \times \frac{2b}{3}\)
\(\implies \frac{a}{b} = \frac{3}{2}\)
Thus, \( a : b : c = 9 : 6 : 4 \).

The Correct Answer is: 9 : 6 : 4