Explanation

(a)
$\frac{x^{n + 1} + y^{n + 1}}{x^{n} + y^{n}} = \sqrt{xy}$
$\Rightarrow x^{n + 1} + y^{n + 1} = (xy)^{\frac{1}{2}}(x^{n} + y^{n})$
$\Rightarrow \ x^{n}.x + y^{n}.y = x^{n}.x^{\frac{1}{2}}y^{\frac{1}{2}} + y^{n}.x^{\frac{1}{2}}y^{\frac{1}{2}}$
$\Rightarrow x^{n}\left( x - \sqrt{xy} \right) + y^{n}\left( y - \sqrt{xy} \right) = 0$
$\Rightarrow x^{n}.\sqrt{x}\left( \sqrt{x} - \sqrt{y} \right) + \sqrt{y}.y^{n}\left( \sqrt{y} - \sqrt{x} \right) = 0$
$\Rightarrow \left( \sqrt{x} - \sqrt{y} \right)\left( x^{n}.\sqrt{x} - y^{n}.\sqrt{y} \right) = 0$
For $x \neq y$
$x^{n}\sqrt{x} = y^{n}\sqrt{y}$
$\Rightarrow \ \left( \frac{x}{y} \right)^{n + \frac{1}{2}} = 1$
$\Rightarrow n + \frac{1}{2} = 0$
$\Rightarrow n = - \frac{1}{2}$