Question

For which value of $n$, expression $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$, will be $A.M$ between $a$ and $b$:

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

Answer 1

The correct option is C $0$

Given the expression $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$, will be $A.M$ between $a,b$.

Then according to the problem,

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

or, $2a^{n+1}+2b^{n+1}=a^{n+1}+a^{n}b+a{b}^n+b^{n+1}$

or, $a^{n+1}+b^{n+1}=a{b}^n+b{a}^n$

or, $a^n(a-b)+b^n(b-a)=0$

or, $(a-b)(a^n-b^n)=0$

or, $a^n=b^n$ for two distinct $a,b$.

Then $n=0$.