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If three positive real numbers $a, b$ and $c(c>a)$ are in Harmonic Progression, then $\log\left ( a+c \right )+\log\left ( a-2b+c \right )$ is equal to:

1. $2\:\log\left ( c-b \right )$
2. $2\:\log\left ( a-c\right )$
3. $2\:\log\left ( c-a\right )$
4. $\log\:a+\log\:b+\log\:c$

harmonic progression is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. Equivalently, it is a sequence of real numbers such that any term in the sequence is the harmonic mean of its two neighbors.

$a,b,c$ are in HP

$\implies \dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}$ are in AP

$\implies \dfrac{1}{b} – \dfrac{1}{a} = \dfrac{1}{c} – \dfrac{1}{b}$

$\implies \dfrac{1}{b} + \dfrac{1}{b} = \dfrac{1}{c} + \dfrac{1}{a}$

$\implies \dfrac{2}{b} = \dfrac{a+c}{ac}$

$\implies \dfrac{b}{2} = \dfrac{ac}{a+c}$

$\implies b = \dfrac{2ac}{a+c}$

$\implies 2b(a+c) = 4ac$

$\log(a+c)+\log(a−2b+c) = \log[(a+c)(a+c-2b)]$

$\implies\log(a+c)+\log(a−2b+c) = \log[(a+c)^{2}-2b(a+c)]$

$\implies\log(a+c)+\log(a−2b+c) = \log[(a+c)^{2}-4ac]$

$\implies\log(a+c)+\log(a−2b+c) = \log(a - c)^{2}$

$\implies\log(a+c)+\log(a−2b+c) = 2\log(c - a)\:\:\:\:(\because c>a)$

So, the correct answer is $(C).$

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