(b) प्रश्नानुसार, $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}} = {(ab)^{1/2}}$

$ \Rightarrow $ ${a^{n + 1}} – {a^{n + 1/2}}{b^{1/2}} + {b^{n + 1}} – {a^{1/2}}{b^{n + 1/2}} = 0$

$ \Rightarrow $ $({a^{n + 1/2}} – {b^{n + 1/2}})({a^{1/2}} – {b^{1/2}}) = 0$

$ \Rightarrow $ ${a^{n + 1/2}} – {b^{n + 1/2}} = 0$

$(\because \;a \ne b$

$\Rightarrow {a^{1/2}} \ne {b^{1/2}})$

$ \Rightarrow $ ${\left( {\frac{a}{b}} \right)^{n + 1/2}} = 1 = {\left( {\frac{a}{b}} \right)^0}$

$\Rightarrow n + \frac{1}{2} = 0 $

$\Rightarrow n =  – \frac{1}{2}$.