First slide

Harmonic Progression

Question

If $\log (a + c) + \log (a + c - 2 b) = 2 \log (a - c)$ then

Moderate

A

$a, b, c a r e i n A . P .$
B

$a, b, c a r e i n G . P$
C

$a, b, c a r e i n H . P$
D

$b + c, c + a, a + b a r e i n H . P$

Solution

Given equation implies
$(a + c) (a + c - 2 b) = (a - c)^{2}$
$\Rightarrow (a + c)^{2} - (a - c)^{2} = 2 b (a + c)$
$\Rightarrow 4 a c = 2 b (a + c)$
$\Rightarrow b = \frac{2 a c}{a + c} \Rightarrow a, b, c$ are in H.P.

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