Three unequal numbers $a$ , $b$ , $c$ are in Arithmetic Progression and $b - a$ , $c - b$ , $a$ are in Geometric Progression.
The value of $\frac{a^{3} + b^{3} + c^{3}}{3 a b c}$ is

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Solution

The correct option is B 2
$(c - b)^{2} = a (b - a)$
i.e., $d^{2} =$ ad where d is the common difference of the A.P.,
$d (d - a) = 0, a n d d \neq 0.$
$\Rightarrow a = d$
$b - a = d \Rightarrow b = a + d = 2 d$
$c = a + 2 d = 3 d$
$∴ \frac{a^{3} + b^{3} + c^{3}}{3 a b c} = 2$

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