Mathematics - Sequence & Series Question with Solution | TestHub

Given expression equal to

$$\begin{array}{rlrlr}& \frac{(a+b)}{c}+\frac{(b+c)}{a}+\frac{(c+a)}{b}& \text{ Or }\frac{a}{c}+\frac{b}{c}+\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}& \text{ Using A.M. }\ge \text{ G.M. }\frac{\frac{a}{c}+\frac{b}{c}+\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}}{6}\ge \sqrt{\frac{a}{c}\frac{b}{c}\frac{b}{a}\frac{c}{a}\frac{c}{b}\frac{a}{b}}& \text{ Or }\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge 6\end{array}$$