Mathematics - Sequence & Series Question with Solution | TestHub

We know that $\mathrm{AM}\ge \mathrm{GM}$...(i) Now, applying Eq. (i) in $a$ and $b$ $\Rightarrow \frac{a+b}{2}\ge \sqrt{ab}$....(ii) applying Eq. (i) in $b$ and $c$ $\Rightarrow \frac{b+c}{2}\ge \sqrt{bc}$....(iii) and applying Eq. (i) in $c$ and $a$ $\Rightarrow \frac{c+a}{2}\ge \sqrt{ca}$....(iv) By E'qs. (ii), (iii) and (iv) $\begin{array}{cc}\Rightarrow & \frac{(a+b)(b+c)(c+a)}{8}\ge abc\\ \Rightarrow & (a+b)(b+c)(c+a)\ge 8abc\end{array}$