Question

If $a,b,c$ are in $H.P.,$ then $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. None

Answer 1

The correct option is C $H.P.$

$a,b,c$ are in H.P.
$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
$\Rightarrow \frac{a+b+c}{a},\frac{a+b+c}{b},\frac{a+b+c}{c}$ are in A.P.
$\because$ Multiplying an A.P. with a constant $(a+b+c),$ resulting sequence will also be in A.P.
$\Rightarrow 1+\frac{b+c}{a},1+\frac{a+c}{b},1+\frac{a+b}{c}$ are in A.P.
$\Rightarrow \frac{b+c}{a},\frac{a+c}{b},\frac{a+b}{c}$ are in A.P.
$\Rightarrow \frac{a}{b+c},\frac{b}{a+c},\frac{c}{a+b}$ are in H.P.