progression and seris

\hspace{-16}$\underline{{\underline{Using $\mathbf{C. S}$ Inequality}}}::\\\\ $\mathbf{(x^2+y^2+z^2).(1^2+2^2+4^2)\geq (x+2y+4z)^2}$\\\\ and equality Hold When $\mathbf{\frac{x}{1}=\frac{y}{2}=\frac{z}{2^2}}$\\\\ So Here $\mathbf{\frac{x}{1}=\frac{y}{2}=\frac{z}{4}=k}$\\\\ i.e $\mathbf{x=k\;,y=2k,z=4k}$\\\\ Means $\mathbf{x\;,y\;,z}$ are in $\mathbf{G.P}$\\\\