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Q.
If $a , b , c$ are distinct positive real in H.P., then the value of the expression, $\frac{ b + a }{ b - a }+\frac{ b + c }{ b - c }$ is equal to
Sequences and Series
Solution:
$ E =\frac{ b + a }{ b - a }+\frac{ b + c }{ b - c }, b =\frac{2 ac }{ a + c }\left( b ^2 \neq ac \right) $
$E =\frac{\left(b^2-b c+a b-a c\right)+\left(b^2+b c-a b-a c\right)}{(b-a)(b-c)}=\frac{2\left(b^2-a c\right)}{b^2-b c-a b+a c}=\frac{2\left(b^2-a c\right)}{b^2-b(a+c)+a c} $
$=\frac{2\left(b^2-a c\right)}{b^2-2 a c+a c}=2$