Question

If $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ be the A.M. of $a$ and $b$, then $n=$

• A. $1$
• B. $-1$
• C. $0$
• D. None of these

Answer 1

The correct option is C $(0$)

If $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ be the A.M. of $a$ and $b$ then,
$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}$
$\Rightarrow 2a^{n+1}+2b^{n+1}=a^{n+1}+b^{n+1}+a^{n}b+a{b}^n$
[Cross multiplying on both sides, we get ]
$\Rightarrow 2a^{n+1}+2b^{n+1}-a^{n+1}-b^{n+1}=a^{n}b+a{b}^n$
$\Rightarrow a^{n+1}+b^{n+1}=a^{n}b+a{b}^n$
$\Rightarrow a^n(a-b)=b^n(a-b)$
$\Rightarrow a^n=b^n$ since $(a\ne b)$
$\therefore n=0$ is only possible solution.