Question

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

• A. -1
• B. 0
• C. 1
• D. None of the above

Answer 1

Answer: (C)

We know that arithmetic mean of $a$ and $b$ is $\frac{a+b}{2}$
$\therefore \frac{a+b}{2}=\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$
$\Rightarrow (a+b)\left(a^{n-1}+b^{n-1}\right)=2\left(a^n+b^n\right)$
$\Rightarrow a^n+\frac{a{b}^n}{b}+\frac{b{a}^n}{a}+b^n=2\left(a^n+b^n\right)$
$\Rightarrow \frac{a{b}^n}{b}+\frac{b{a}^n}{a}=a^n+b^n$
$\Rightarrow a^n\left(\frac{a-b}{a}\right)=-b^n\left(\frac{b-a}{b}\right)$
$\Rightarrow \frac{a^n}{b^n}=\frac{a}{b}$
$\Rightarrow {\left(\frac{a}{b}\right)}^n={\left(\frac{a}{b}\right)}^1$
$\therefore n=1$