Question

If $a^2,b^2,c^2$ are in A.P. then the following are also in A.P. True or False? If true write 1 otherwise write 0 $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}.$

Answer 1

Assume $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}.$ are in A.P

$\Rightarrow \frac{b}{c+a}-\frac{a}{b+c}=\frac{c}{a+b}-\frac{b}{c+a}$

$\Rightarrow (\frac{b}{c+a}+1)-(\frac{a}{b+c}+1)=(\frac{c}{a+b}+1)-(\frac{b}{c+a}+1)$

$\Rightarrow \frac{1}{c+a}-\frac{1}{b+c}=\frac{1}{a+b}-\frac{1}{c+a}$

$\Rightarrow \frac{b-a}{b+c}=\frac{c-b}{a+b}$

$\Rightarrow b^2-a^2=c^2-b^2\Rightarrow 2b^2=a^2+c^2$
Hence our assumption is True.
Thus answer is 1.