Arithmetic mean between $a$ and $b$ is given by $\frac{a+b}{2}$
$$
\begin{aligned}
&\therefore \quad \frac{a+b}{2}=\frac{a^{n+1}+b^{n+1}}{a^n+b^n} \\
&\Rightarrow 2 a^{n+1}+2 b^{n+1}=a^{n+1}+a^n b+b^n a+b^{n+1} \\
&\Rightarrow\left(a^{n+1}-a^n b\right)+\left(b^{n+1}-a b^n\right)=0 \\
&\Rightarrow a^n(a-b)+b^n(b-a)=0 \\
&\Rightarrow\left(a^n-b^n\right)(a-b)=0 \\
&\Rightarrow a^n-b^n=0 \quad(\because a-b \neq 0) \\
&\Rightarrow\left(\frac{a}{b}\right)^n=1 \Rightarrow\left(\frac{a}{b}\right)^n=\left(\frac{a}{b}\right)^0 \\
&\Rightarrow n=0
\end{aligned}
$$