Using the properties of determinants, prove that : $\begin{bmatrix} x-3 & x-4 & x-\alpha \\ x-2 & x-3 & x-\beta \\ x-1 & x-2 & x-\gamma \end{bmatrix} = 0, \: where\: \alpha, \beta, \gamma \: are \: in \ A.P.$

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