Question

If $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the AM between a and b. Then, find the value of n.

Answer 1

AM b/w a and b = $\frac{a+b}{2}$

Therefore,

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

$2(a^{n+1}+b^{n+1})=(a^n+b^n)(a+b)$

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

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

$a^{n+1}-a^{n}b=a{b}^n-b^{n+1}$

$a^n=b^n$

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

$(\frac{a}{b}{)}^n=(\frac{a}{b}{)}^0$

$n=0$