$\text{ If }\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\text{ and }\mathrm{p}\text{ are distinct non }$$\text{ zero real numbers such that }$$\left(a^2+b^2+c^2\right)p^2-2(ab+bc+cd)p$$+\left(b^2+c^2+d^2\right)\le 0\text{ then }a,b,c,d\text{ are }$in

$\begin{array}{c}\begin{array}{r}\left(a^2+b^2+c^2\right)p^2-2(ab+bc+cd)p\\ +b^2+c^2+d^2\le 0\end{array}\\ \begin{array}{r}\Rightarrow \left(a^2{p}^2-2abp+b^2\right)+\left(b^2{p}^2-2bcp+c^2\right)\\ +\left(c^2{p}^2-2cdp+d^2\right)\le 0\end{array}\end{array}$

   $\Rightarrow {\left(ab-p\right)}^2+{\left(bp-c\right)}^2+{\left(cp-d\right)}^2\le 0$

$\begin{array}{c}\Rightarrow ap-b=0,bp-c=0andcp-d=0\\ \Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}\Rightarrow a,b,c\text{ and }d\text{ are in GP}\\ \text{ Also ad }=bc\end{array}$