If _e a , _ e b , _ e c are in an A.P. and _e a - _e 2 b, _e 2 b-… - Vidyadip

MCQ

If $\log _e a , \log _{ e } b , \log _{ e } c$ are in an $A.P.$ and $\log _e a -$
$\log _e 2 b, \log _e 2 b-\log _e 3 c , \log _e 3 c -\log _{ e } a$ 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

$\log _{ e } a , \log _{ e } b , \log _{ e } c$ are in $A.P.$
$\therefore b^2=ac ...(i)$
Also
$\log _{ c }\left(\frac{ a }{2 b}\right), \log _{ e }\left(\frac{2 b}{3 c }\right), \log _{ e }\left(\frac{3 c }{ a }\right)$ are in $A.P.$
$\left(\frac{2 b}{3 c}\right)^2=\frac{a}{2 b} \times \frac{3 c}{a}$
$\frac{b}{c}=\frac{3}{2}$
Putting in eq. $(i) b^2= a \times \frac{2 b}{3}$
$\frac{a}{b}=\frac{3}{2}$
$a: b: c=9: 6: 4$

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