Question

If $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ represents arithmetic mean, geometric mean, harmonic mean between two distinct positive numbers $a$ and $b$ for $n=\alpha ,\beta ,\gamma$ respectively, then the equation whose roots are $\alpha ,2\beta ,\gamma$ is

• A. $2x^3+3x^2+x=0$
• B. $x^3+2x^2+3x=0$
• C. $x^3+2x^2+x=0$
• D. $x^3-2x^2+3x=0$

Answer 1

The correct option is C $x^3+2x^2+x=0$

$\frac{a^{\alpha +1}+b^{\alpha +1}}{a^{\alpha }+b^{\alpha }}=\frac{a+b}{2}\Rightarrow \alpha =0$

$\frac{a^{\beta +1}+b^{\beta +1}}{a^{\beta }+b^{\beta }}=\sqrt{ab}$
$\Rightarrow a^{\beta +1}+b^{\beta +1}=a^{\beta +1/2}{b}^{1/2}+b^{\beta +1/2}{a}^{1/2}$
$\Rightarrow a^{\beta +1/2}\left(a^{1/2}{b}^{1/2}\right)-b^{\beta +1/2}\left(a^{1/2}{b}^{1/2}\right)=0$
$\Rightarrow a^{\beta +1/2}-b^{\beta +1/2}=0$
$\Rightarrow a^{\beta +1/2}=b^{\beta +1/2}$
$\Rightarrow \beta =-\frac{1}{2}$

$\frac{a^{\gamma +1}+b^{\gamma +1}}{a^{\gamma }+b^{\gamma }}=\frac{2ab}{a+b}\Rightarrow \gamma =-1$

The equation whose roots are $\alpha ,2\beta ,\gamma$ is
$(x-\alpha )(x-2\beta )(x-\gamma )=0$
$\Rightarrow x(x+1)(x+1)=0$
$\Rightarrow x^3+2x^2+x=0$