Explanation

(c)
We know that, arithmetic mean of $a$ and $b = \frac{a + b}{2}.$
But given that $\frac{a + b}{2} = \frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$
$\Rightarrow \ a^{n} + b^{n} + \frac{ab^{n}}{b} + \frac{ba^{n}}{a} = 2(a^{n} + b^{n})$
$\Rightarrow \ \frac{a}{b}b^{n} + \frac{b}{a}a^{n} = a^{n} + b^{n}$
$\Rightarrow a^{n}\left( \frac{a - b}{a} \right) = - b^{n}\left( \frac{b - a}{b} \right)$
$\Rightarrow \ \ \left( \ \frac{a}{b}\ \right)^{n} = \left( \ \frac{a}{b}\ \right)$
$\therefore\ n = 1$