Question

Given $S=\frac{a^{n+1}+b^{n+1}}{a^n+b^n},a,b>0$
$\left(1\right)S=$ A.M. of $a$ and $b$, if $n=0$
$\left(2\right)S=$ G.M. of $a$ and $b$, if $n=\frac{1}{2}$
$\left(3\right)S=$ H.M. of $a$ and $b$, if $n=-1$, then

• A. only $\left(1\right)$ is correct
• B. both $\left(1\right)$ and $\left(2\right)$ are correct
• C. both $\left(1\right)$ and $\left(3\right)$ are correct
• D. all are correct

Answer 1

The correct option is C both $\left(1\right)$ and $\left(3\right)$ are correct

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

$n=0\Rightarrow S=\frac{a+b}{2}$

$n=\frac{1}{2}\Rightarrow S=\frac{a^{3/2}+b^{3/2}}{a^{1/2}+b^{1/2}}\ne \sqrt{ab}$

$n=-1\Rightarrow S=\frac{2ab}{a+b}$