Question

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

Answer 1

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}+H.M.ofaandb=\frac{2ab}{a+b}$
$\therefore a.a^{n+1}+a{b}^{n+1}+b{a}^{n+1}+b.b^{n+1}$
$=2a^{n+1}b+2b^{n+1}a$
$\therefore a.a^{n+1}+b.b^{n+1}=a^{n+1}b+b^{n+1}a$
$or{a}^{n+1}(a-b)=b^{n+1}(a-b)$
$or{a}^{n+1}=b^{n+1}$
$or{\left(\frac{a}{b}\right)}^{n+1}=1\therefore n+1=0orn=-1$