Question

If a,b,c be in H.P. prove that
$\frac{1}{a}+\frac{1}{b+c},\frac{1}{b}+\frac{1}{c+a},\frac{1}{c}+\frac{a}{a+b}$ are in H.P.

Answer 1

We have to prove that
$\frac{a+b+c}{a(b+c)},\frac{a+b+c}{b(c+a)},\frac{a+b+c}{c(a+b)}$ to be H.P.
Taking reciprocal and cancel a + b + c.
Thus we have to prove that
a(b + c), b(c + a), c(a + b) to be in A.P.
or (ab + bc + ca) - bc, $\sum$ ab - ca, $\sum$ ab - ab to be in A.P.
or -bc, -ca, -ab to be in A.P. Divide by - abc
or $\frac{1}{a},\frac{1}{b},\frac{1}{c},$ to be in A.P.
or a,b,c to be in H.P., which is true.