Question

The arithmetic mean of a and b is $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$. The value of n is

• A. -1
• B. \N
• C. 1
• D. a

Answer 1

The correct option is C 1

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