If the expansion of ${\left(\frac{{\displaystyle 3\sqrt{x}}}{{\displaystyle 7}}-\frac{{\displaystyle 5}}{{\displaystyle 2x\sqrt{x}}}\right)}^{13n}$ contains a term independent of  $x$ in 14^th  term $n$ should be

$T_{14}{=}^{13n}{C}_{13}{\left(\frac{{\displaystyle 3}}{{\displaystyle 7}}{x}^{\frac{{\displaystyle 1}}{{\displaystyle 2}}}\right)}^{3n-13}{(-1)}^{13}{\left(\frac{{\displaystyle 5}}{{\displaystyle 2}}{x}^{\frac{{\displaystyle -3}}{{\displaystyle 2}}}\right)}^{13}$

${=}^{13n}{C}_{13}{\left(\frac{{\displaystyle 3}}{{\displaystyle 7}}\right)}^{13n-13}{(-1)}^{13}{\left(\frac{{\displaystyle 5}}{{\displaystyle 2}}\right)}^{13}{x}^{\frac{13n-13}{2}}{x}^{\frac{39}{2}}$

$={}^{13n}{C}_{13}{\left(\frac{{\displaystyle 3}}{{\displaystyle 7}}\right)}^{13n-13}{(-1)}^{13}\cdot x^{\frac{13n-13-39}{2}}$

For this term to be independent of $x$, we get

$\begin{array}{l}13n-52=0\\ \Rightarrow n=4\end{array}$