The G.M. of the numbers  $3,3^2,3^3...,3^n$ is 

Clearly,  

    Required GM   $={\left(3\times 3^2\times 3^3\times \dots \times 3^n\right)}^{1/n}$

 $\Rightarrow$ Required GM  =  $= {\left\{3^{1+2+\dots +n}\right\}}^{1/n}={\left[3^{\frac{n(n+1)}{2}}\right\}}^{1/n}=3^{\frac{n+1}{2}}$