Question

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

• A. 0
• B. 1
• C. -1
• D. none of these

Answer 1

The correct option is A 0

The A.M. of a and b is $\frac{a+b}{2}$.

Therefore, $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ will be the A.M. of a and b, if

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

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

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

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

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

$\Rightarrow a^n=b^n$

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

$\Rightarrow (\frac{a}{b}{)}^n=1$

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

$\Rightarrow n=0$