Question

$f(x)=0$ is a cubic equation with positive and distinct roots $\alpha ,\beta ,\gamma$ such that $\beta$ is harmonic mean between the roots of $f^{\prime }(x)=0$. If $r=\left[\frac{2\beta }{\alpha +\gamma }\right]+\left[\frac{2\alpha \gamma }{\alpha \beta +\beta \gamma }\right]$, then the value of $\sum _{i=1}^3(i{)}^r$ is
$[$ Note $:[k]$ denotes greatest integer function less than or equal to $k$. $]$

Answer 1

Answer: 3

$\beta =\frac{2x_1{x}_2}{x_1+x_2}=\frac{2b}{-2a}=\frac{b}{-a}=\frac{-(\alpha \beta +\beta \gamma +\gamma \alpha )}{-(\alpha +\beta +\gamma )}$
$\alpha \beta +{\beta }^2+\beta \gamma =\alpha \beta +\beta \gamma +\gamma \alpha$
${\beta }^2=\alpha \gamma \Rightarrow \alpha ,\beta ,\gamma \text{ are in G.P. }$
$r=\left[\frac{G\cdot M}{\text{ A.M. }}\right]+\left[\frac{\text{ H.M. }}{\text{ G.M. }}\right]=0+0=0$
$\sum _{i=I}^{3}1=1+1+1=3$