If ln(a+c) , ln(a-c) , ln(a-2b+c) are in A.P, then

a,b,c are in A.P.

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$a^{2}, b^{2}, c^{2}$ are in A.P.

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a,b,c are in G.P.

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a,b,c are in H.P.

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Solution

The correct option is D
a,b,c are in H.P.

$I n (a + c), l n (a - c), l n (a - 2 b + c) a r e i n A . P . \Rightarrow a + c, a - c, a - 2 b + c a r e i n G . P . \Rightarrow (c - a)^{2} = (a + c) (a - 2 b + c) \Rightarrow (c - a)^{2} = (a + c)^{2} - 2 b (a + c) \Rightarrow 2 b (a + c) = (a + c)^{2} - (c - a)^{2} \Rightarrow 2 b (a + c) = 4 a c \Rightarrow b = \frac{2 a c}{a + c}$
a,b,c are in H.P.

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