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