Question

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

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

Answer 1

The correct option is C $\frac{1}{2}$

Given $\frac{a^n+b^n}{a^n+b^{n-1}}=\sqrt{ab}$

$\frac{a^n+b^n}{\sqrt{ab}}=a^{n-1}+b^{n-1}$

$a^{n-1/2}/\sqrt{b}+b^{n-1/2}/\sqrt{a}-a^{n-1}+b^{n-1}$

$\Rightarrow a^{n-1/2}{b}^{-1/2}-a^{n-1}+b^{n-1/2}{a}^{-1/2}-b^{n-1}=0$

$\Rightarrow a^{n-1/2}\left(b^{-1/2}-a^{-1/2}\right)+b^{n-1/2}\left(a^{-1/2}-b^{-1/2}\right)=0$

$\Rightarrow \left(b^{-1/2}-a^{-1/2}\right)\left(a^{n-1/2}-b^{n-1/2}\right)=0$

$\Rightarrow b^{-1/2}=a^{-1/2}$

Or $\sqrt{a}=\sqrt{b}$

Or $a^{n-1/2}=b^{n-1/2}$

${\left(\frac{a}{b}\right)}^{n-1/2}=1={\left(\frac{a}{b}\right)}^0$

$n-\frac{1}{2}=0$

$n=\frac{1}{2}$