Question

If 1/a + 1/c + 1/a-b + 1/c-b = 0. prove that a , b/2 , c are in A.P.​

Answer 1

Answer:

After rearranging we get

\Rightarrow \frac{1}{a}+\frac{1}{c-b}+\frac{1}{a-b}+\frac{1}{c}=0

After taking LCM of first two and last to terms we get

\Rightarrow \frac{c-b+a}{a(c-b)}+\frac{c+a-b}{c(a-b)}=0

since a+c-b is not equal to 0

therefore \frac{1}{a(c-b)}+\frac{1}{c(a-b)}=0

Taking LCM

\Rightarrow \frac{ac-bc+ac-ab}{ac(c-b)(a-b)}=0

Therefore numerator is equal to zero

\Rightarrow \ ac-bc+ac-ab = 0

\Rightarrow \ 2ac = bc+ab

\Rightarrow \ \frac{2}{b} = \frac{1}{a}+\frac{1}{c}

Which is necessory condition to proove that a, b and c are in HP