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If $a, b, c$ are in HP, then $\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ will be in
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The correct answer is:
$\mathrm{HP}$
$$
\begin{aligned}
\mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in HP. } & \\
\Rightarrow \frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{a}}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{b}}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & 1+\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}, 1+\frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}}, 1+\frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}, \frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}}, \frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}, \frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}, \frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}} \text { are in HP. }
\end{aligned}
$$
\begin{aligned}
\mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in HP. } & \\
\Rightarrow \frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{a}}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{b}}, \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & 1+\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}, 1+\frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}}, 1+\frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}, \frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}}, \frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}} \text { are in AP. } \\
\Rightarrow & \frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}, \frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}, \frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}} \text { are in HP. }
\end{aligned}
$$
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