Question

If the harmonic mean between $a$ and $b$ is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$, then $n=$

• A. $0$
• B. $-1$
• C. $\frac{-1}{2}$
• D. $1$

Answer 1

The correct option is B $-1$

$(a+b)\left[\right(a{)}^{n+1}+\left(b{)}^{n+1}\right]=\left(2ab\right)\left(\right(a{)}^n+\left(b{)}^n\right))$
$(a{)}^{n+2}+(b{)}^{n+2}+b(a{)}^{n+1}+a(b{)}^{n+1}=2(a{)}^{n+1}b+2a(b{)}^{n+1}$
$(a{)}^{n+2}+(b{)}^{n+2}-b(a{)}^{n+1}-a(b{)}^{n+1}=0$
$\left[\right(a{)}^{n+1}-\left(b{)}^{n+1}\right](a-b)=0$

$(a{)}^{n+1}=(b{)}^{n+1}$ this is only possible when $n+1=0,n=-1$