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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 $n$.
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A. M. between $a$ and $b=\frac{a+b}{2}$
$\begin{aligned} \therefore \frac{a^n+b^n}{a^{n-1}+b^{n-1}} &=\frac{a+b}{2} \\ 2 a^n+2 b^n &=a^n+a b^{n-1}+a^{n-1} b+b^n \end{aligned}$
$\Rightarrow \quad a^n a^{n-1} b-a b^{n-1}+b^n=0$
$\Rightarrow \quad a^{n-1}(a-b)-b^{n-1}(a-b)=0$
$\Rightarrow \quad(a-b)\left(a^n-b^{n-1}\right)+b^n=0 \quad[\because a \neq 0]$
$\Rightarrow \quad a^{n-1}-b^{n-1}=0 \Rightarrow a^{n-1}=b^{n-1}$
$\left(\frac{a}{b}\right)^{n-1}=1=\left(\frac{a}{b}\right)^0 \Rightarrow n-1=0 \Rightarrow n=1$
$\begin{aligned} \therefore \frac{a^n+b^n}{a^{n-1}+b^{n-1}} &=\frac{a+b}{2} \\ 2 a^n+2 b^n &=a^n+a b^{n-1}+a^{n-1} b+b^n \end{aligned}$
$\Rightarrow \quad a^n a^{n-1} b-a b^{n-1}+b^n=0$
$\Rightarrow \quad a^{n-1}(a-b)-b^{n-1}(a-b)=0$
$\Rightarrow \quad(a-b)\left(a^n-b^{n-1}\right)+b^n=0 \quad[\because a \neq 0]$
$\Rightarrow \quad a^{n-1}-b^{n-1}=0 \Rightarrow a^{n-1}=b^{n-1}$
$\left(\frac{a}{b}\right)^{n-1}=1=\left(\frac{a}{b}\right)^0 \Rightarrow n-1=0 \Rightarrow n=1$
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