Question

For what value of $n,\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the geometric mean of $a$ and $b$ ?

Answer 1

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\sqrt{ab}=a^{1/2}{b}^{1/2}$ by given condition
$\therefore a^{n+1}+b^{n+1}=a^n{a}^{1/2}{b}^{1/2}+b^n{b}^{1/2}{a}^{1/2}=a^{n+1/2}.\sqrt{b}+b^{n+1/2}\sqrt{a}$
$\therefore a^{n+1/2}(\sqrt{a}-\sqrt{b})=b^{n+1/2}(\sqrt{a}-\sqrt{b})$
$\therefore a^{n+1/2}=b^{n+1/2}$
Above is possible only when $n+\frac{1}{2}=0$
$\therefore$ $n=-\frac{1}{2}$.