If _e a, _e b, _e c are in A.P., and _e a - _e 2b, _e 2b - _e 3c, _e 3c - _e a are also in…

Question

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

Accepted Answer

$\log a,\log b,\log c$ in A.P. $\Rightarrow b^2=ac$.

Let $X_1=\ln a-\ln2b$, $X_2=\ln2b-\ln3c$, $X_3=\ln3c-\ln a$; note $X_1+X_2+X_3=0$ always. If they are in A.P., $X_1+X_3=2X_2$, and combined with the sum being zero, $X_2=0 \Rightarrow 2b=3c \Rightarrow c=\dfrac{2b}{3}$.

From $b^2=ac = a\cdot\dfrac{2b}{3} \Rightarrow a = \dfrac{3b}{2}$.

$$a:b:c = \frac{3b}{2}:b:\frac{2b}{3} = 9:6:4 \quad(\text{multiplying by }6).$$

Solutions · 1