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

Answer: (A)

9 : 6 : 4

Answer 2

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