If [ _e a, _e ~b, _e c] are in an A.P.

Question

If $$\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$$ are in an A.P. and $$\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e$$ a are also in an A.P, then $a: b: c$ is equal to

Accepted Answer

$$\log _{\mathrm{e}} \mathrm{a}, \log _{\mathrm{e}} \mathrm{b}, \log _{\mathrm{e}} \mathrm{c}$$ 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) $$b^2=a \times \frac{2 b}{3}$$

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

Solutions · 1