If the geometric mean between a and b is frac | vedclass

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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-mu

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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: $a^{n+1} - a^{n+1/2} b^{1/2} = a^{1/2} b^{n+1/2} - b^{n+1}$.$a^{n+1/2} (a^{1/2} - b^{1/2}) = b^{n+1/2} (a^{1/2} - b^{1/2})$.Assuming $a 
eq b$,we can divide by $(a^{1/2} - b^{1/2})$:$a^{n+1/2} = b^{n+1/2}$.$(a/b)^{n+1/2} = 1 = (a/b)^0$.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 what is the value of $n$?

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