QJEEM24Ma9731: JEE Main 2024 Mathematics - Class 11 JEEM24Ma9731 Solution: If are in an A.P. and a are also in an A.P, then is equal to

$\log _{\mathrm{e}} \mathrm{a}, \log _{\mathrm{e}} \mathrm{b}, \log _{\mathrm{e}} \mathrm{c}$ are in A.P.

$\therefore \mathrm{b}^2=\mathrm{ac}$ ..... (i)

Also

$\begin{aligned}& \log _e\left(\frac{a}{2 b}\right), \log _e\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right) \text { are in A.P. } \\& \left(\frac{2 b}{3 c}\right)^2=\frac{a}{2 b} \times \frac{3 c}{a} \\& \frac{b}{c}=\frac{3}{2}\end{aligned}$

Putting in eq. (i) $b^2=a \times \frac{2 b}{3}$

$\begin{aligned}& \frac{\mathrm{a}}{\mathrm{b}}=\frac{3}{2} \\& \mathrm{a}: \mathrm{b}: \mathrm{c}=9: 6: 4\end{aligned}$