To prove $x,y,z$ are in G.P. we have to prove that $\large\frac{y}{x}=\frac{z}{y}$

or we should prove that $y^2=xz$

Given that in a G.P. $t_4=x,\:\:t_{10}=y,\:\:t_{16}=z$

$\Rightarrow\:a.r^{4-1}=a.r^3=x$

$a.r^{10-1}=a.r^9=y\:\:\:and\:\: a.r^{16-1}=a.r^{15}=z$

$\Rightarrow\:xz=(a.r^3).(a.r^{15})$

$\Rightarrow\:xz=a^2.r^{15+3}=a^2.r^{18}$......(i)

and

$y^2=(a.r^9)^2=a^2.r^{18}$.....(ii)

From (i) and (ii) $ y^2=xz$

$\therefore\:x,y,z$ are in G.P.

Hence Proved.