Question

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

Answer 1

According to the given condition,

Given is the A.M between a and b, therefore,

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

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

Cross multiply.

$a^n\times a+a^n\times b+b^n\times a+b^n\times b=2a^n\times a+2b^n\times b$

$a^{n}b+b^{n}a=a^{n}a+b^{n}b$

$a^{n}b-a^{n}a=b^{n}b-b^{n}a$

$a^n(b-a)=b^n(b-a)$

$a^n=b^n$

a and b can only be equal if n=0. ($x^n=1$)

Thus, $n=0.$