The value of $\text{'}n\text{'}$  such that  $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the G.M between $a$  and $b$  is

Put $n=-1$

$\frac{a^0+b^0}{a^{-1}+b^{-1}}=\frac{2}{\left(\frac{a+b}{ab}\right)}=\frac{2ab}{a+b}$    $\left(H.M\right)$

Put $n=0$

$\frac{a+b}{a^0+b^0}=\frac{a+b}{2}$ $\left(A.M\right)$

Put $n=\frac{1}{2}$

$\frac{a^{\frac{1}{2}}+b^{\frac{1}{2}}}{\frac{1}{a^{\frac{1}{2}}}+\frac{1}{b^{\frac{1}{2}}}}=a^{\frac{1}{2}}.b^{\frac{1}{2}}={\left(ab\right)}^{\frac{1}{2}}=\sqrt{ab}$  $\left(G.M\right)$