Question

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

Answer 1

We have,
A.M. between $a$ and $b=\frac{a+b}{2}$
It is given that
$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the A.M. between $a$ and $b$.
$\therefore \frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}$
$⟹2(a^{n+1}+b^{n+1})=(a^n+b^2)(a+b)$

$⟹2a^{n+1}+b^{n+1}=a^{n+1}+b^{n+1}+a^{n}b+b^{n+1}$

$⟹a^{n+1}+b^{n+1}=a{b}^n+a^{n}b$

$⟹a^{n+1}-a^{n}b=a{b}^n-b^{n+1}$

$⟹a^n(a-b)=b^n(a-b)$

$⟹a^n=b^n$

$⟹\frac{a^n}{b^n}=1⟹{\left(\frac{a}{b}\right)}^n={\left(\frac{a}{b}\right)}^0⟹n=0$