Question

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

Answer 1

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=$ H.M. of a and b $=\frac{2ab}{a+b}$

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

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

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

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

or ${\left(\frac{a}{b}\right)}^{n+1}=1$

$\therefore n+1=0orn=-1$