Guess and check?

Note LHS denotes the Sum of a Geometric Progression with finite terms which equals $\dfrac{\sqrt 6((\sqrt 6)^n-1)}{\sqrt 6-1}=\dfrac{\sqrt 6( 6^{n/2}-1)}{\sqrt 6-1}$

And according to question:-

$\dfrac{\sqrt 6(6^{n/2}-1)}{\sqrt 6-1}=9331\sqrt 6+9330$

$\implies 6^{n/2}-1= \dfrac{(9331\sqrt 6+9330)(\sqrt 6-1)}{\sqrt 6}$

$\implies 6^{n/2}-\cancel 1=7776\sqrt 6-\cancel 1$

$6^{n/2}=7776\sqrt 6=6^{5+\frac 12}=6^{11/2}$

$\implies \boxed{n=11}$