Question

For what value of $n,\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmetic mean of a and b?

Answer 1

By the given condition $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}$

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

or $a^{n+1}-a^{n}b=b^{n}a-b^{n+1}$ or $a^n(a-b)=b^n(a-b)$

$\therefore a^n=b^n.$

Above is possible only when $n=0\because a^0=b^0=1.$