Question

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

Answer 1

We kinow that A.M. between a and b is $\frac{a+b}{2}$.

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

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

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

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

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

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

$\Rightarrow {\left(\frac{a}{b}\right)}^{n-1}=\frac{a^{\circ }}{b^{\circ }}\Rightarrow n-1=0\Rightarrow n-1.$