Question

Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the geometric means between a and b.

Answer 1

$G.M$. of $a,b=\sqrt{ab}$.

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\sqrt{ab}$

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

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

If $a=b$ then

$n$ can be anything.