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Q.
Let $\alpha, \beta, \gamma$ be the roots of the equation $x^3+3 a x^2+3 b x+c-\hat{0}$. If $\alpha, \beta, \gamma$ are in H.P. then $\beta$ is equal to
Sequences and Series
Solution:
$\alpha+\beta+\gamma=-3 a ,\,\,\,\, \alpha \beta \gamma=- c $
$ \alpha \beta+\beta \gamma+\gamma \alpha=3 b \Rightarrow \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{3 b }{\alpha \beta \gamma}$
$ \Rightarrow \frac{2}{\beta}+\frac{1}{\beta}=\frac{3 b }{- c } (\because \alpha, \beta, \gamma \text { in H.P }) $
$\Rightarrow \frac{1}{\beta}=-\frac{ b }{ c } \Rightarrow \beta=\frac{- c }{ b }$