Question

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

• A. $a,b,c$ are in AP.
• B. $a^2,b^2,c^2$ are in AP.
• C. $a,b,c$ are in GP.
• D. $a,b,c$ are in HP.

Answer 1

The correct option is D

$a,b,c$ are in HP.

Explanation for the correct option.

It is given than $\ln (a+c),\ln (a-c),\ln (a-2b+c)$ are in AP,

$\begin{array}{rcl}\Rightarrow 2\ln \left(a-c\right)& =& \ln \left(a+c\right)+\ln \left(a-2b+c\right)\\ \Rightarrow {\left(a-c\right)}^2& =& \left(a+c\right)\left(a-2b+c\right)\\ & \Rightarrow & a^2+c^2-2ac=a^2-2ab+2ac-2bc+c^2\\ \Rightarrow 4ac& =& 2ab+2bc\\ \Rightarrow 2ac& =& ab+bc\end{array}$

Dividing both sides by $abc$, we get

$\frac{2}{b}=\frac{1}{c}+\frac{1}{a}$

Therefore, $a,b,c$ are in HP.

Hence, option D is correct.