Question

If the geometric mean between $a$ and $b$ is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$, then the value of $n$ is.

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

Answer 1

The correct option is B $\frac{-1}{2}$

If $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ be the G.M. of a and b then,
$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=(ab{)}^{1/2}$
$\Rightarrow a^{n+1}+b^{n+1}=a^{n+1/2}{b}^{1/2}+a^{1/2}{b}^{n+1/2}$
$\Rightarrow a^{n+1/2}(a^{1/2}-b^{1/2})=b^{n+1/2}(a^{1/2}-b^{1/2})\Rightarrow a^{n+1/2}=b^{n+1/2}$ since $(a\ne b)$
$\therefore n=-1/2$ is only possible solution.