Question

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

• A. $a,b,c$ are in A.P.
• B. $a^2,b^2,c^2$ are in A.P.
• C. $a,b,c$ are in G.P.
• D. $a,b,c$ are in H.P.

Answer 1

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

$ln(a+c),ln(c-a),ln(a-2b+c)$ are in A.P
$\Rightarrow 2ln(c-a)=ln(a+c)+ln(a-2b+c)$herefore
$\Rightarrow (c-a{)}^2=(a+c\left)\right(a-2b+c)$
$\Rightarrow -2ac=-2ab+2ac-2bc$
$\Rightarrow 2ac=ab+bc$
$\therefore \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$
i.e $a,b,c$ are in HP.
Hence, option D.