$\frac{{{a}^{n\,+\,1}}\,+\,{{b}^{n\,+\,1}}}{{{a}^{n}}\,+\,\,{{b}^{n}}}\,\,=\,\,\frac{a\,\,+\,\,b}{2}$

$\Rightarrow \,\,{{a}^{n\,+\,1}}\,-\,\,a{{b}^{n}}\,+\,\,{{b}^{n\,+\,1}}\,-\,\,b{{a}^{n}}\,=\,\,0\,\,$

$\Rightarrow \,\,(a\,\,-\,\,b)\,\,({{a}^{n}}\,-\,\,{{b}^{n}})\,\,=\,\,0$

જો $\,{{a}^{n}}\,-\,\,{{b}^{n}}\,=\,\,0\,$ હોય, તો ${{\left( \frac{\text{a}}{\text{b}} \right)}^{n}}\,=\,\,1\,\,=\,\,{{\left( \frac{a}{b} \right)}^{0}}\,\,$

$\Rightarrow \,\,n\,\,=\,\,0$