If is the A.M. between a and b, then find the value of n.


If $\frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}$ is the A.M. between a and b, then find the value of n.


A.M. of $a$ and $b=\frac{a+b}{2}$ 

According to the given condition,



$\Rightarrow a^{n}+a b^{n-1}+b a^{n-1}+b^{n}=2 a^{n}+2 b^{n}$

$\Rightarrow a b^{n-1}+a^{n-1} b=a^{n}+b^{n}$

$\Rightarrow a b^{n-1}-b^{n}=a^{n}-a^{n-1} b$

$\Rightarrow b^{n-1}(a-b)=a^{n-1}(a-b)$

$\Rightarrow b^{n-1}=a^{n-1}$


$\Rightarrow n-1=0$

$\Rightarrow n=1$



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