Question

If $\frac{b+c-a}{a}$ , $\frac{c+a-b}{b}$ , $\frac{a+b-c}{c}$ are in AP, then which of the following is in AP?

• A. $a$ , $b$ , $c$
• B. $a^2$ , $b^2$ , $c^2$
• C. $\frac{1}{a}$ , $\frac{1}{b}$ , $\frac{1}{c}$
• D. none of these

Answer 1

The correct option is C $\frac{1}{a}$ , $\frac{1}{b}$ , $\frac{1}{c}$

Since, $\frac{b+c-a}{a}$ , $\frac{c+a-b}{b}$ , $\frac{a+b-c}{c}$ are in A.P.
$\frac{b+c-a}{a}+1$ , $\frac{c+a-b}{b}+1$ , $\frac{a+b-c}{c}+1$ are in A.P.
$\Rightarrow \frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in $AP$
$\Rightarrow \frac{b+c}{a}+1,\frac{c+a}{b}+1,\frac{a+b}{c}+1$ are in $AP$
$\Rightarrow \frac{a+b+c}{a},\frac{a+b+c}{b},\frac{a+b+c}{c}$ are in $AP$
Divide by $a+b+c$

$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in $AP$