Question

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

• A. $0$
• B. $1$
• C. $1/2$
• D. $noneofthese$

Answer 1

Answer: (C)

$1/2$

We have,

$\frac{a^n+b^n}{a^{n-1}+b^{n-1}}=a^{\frac{1}{2}}{b}^{\frac{1}{2}}$

$a^n+b^n=a^{n-\frac{1}{2}}{b}^{\frac{1}{2}}+a^{\frac{1}{2}}+b^{n-\frac{1}{2}}$

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

$a^{n-\frac{1}{2}}-b^{n-\frac{1}{2}}=0$

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

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

Hence, the option $\left(C\right)$ is the correct answer.