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(B) The geometric mean between $a$ and $b$ is given by $\sqrt{ab} = (ab)^{1/2}$.Given that $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n} = (ab)^{1/2}$.Cros

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$-1/2$

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(B) The geometric mean between $a$ and $b$ is given by $\sqrt{ab} = (ab)^{1/2}$.Given that $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n} = (ab)^{1/2}$.Cross-multiplying,we get $a^{n + 1} + b^{n + 1} = (ab)^{1/2}(a^n + b^n)$.$a^{n + 1} + b^{n + 1} = a^{n + 1/2}b^{1/2} + a^{1/2}b^{n + 1/2}$.Rearranging the terms: $a^{n + 1} - a^{n + 1/2}b^{1/2} + b^{n + 1} - a^{1/2}b^{n + 1/2} = 0$.$a^{n + 1/2}(a^{1/2} - b^{1/2}) - b^{n + 1/2}(a^{1/2} - b^{1/2}) = 0$.$(a^{n + 1/2} - b^{n + 1/2})(a^{1/2} - b^{1/2}) = 0$.Since $a 
eq b$,$a^{1/2} - b^{1/2} 
eq 0$,so we must have $a^{n + 1/2} - b^{n + 1/2} = 0$.$a^{n + 1/2} = b^{n + 1/2}$.$(a/b)^{n + 1/2} = 1 = (a/b)^0$.Equating the exponents,$n + 1/2 = 0$,which gives $n = -1/2$.

If the geometric mean between $a$ and $b$ is $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$,then the value of $n$ is

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