\(\displaystyle{-2)-2)-2)-2}\)

Algebra Level 5

The coefficient of \(\displaystyle{x^2}\) appearing after parenthesis have been removed and like terms have been collected in \[\underbrace{\Big( ..............(x-2)^{2} -2)^{2} - 2)^{2}..)......\Big)^{2} }_\text{k times} \] can be expressed as \[\dfrac{\alpha^{2k-1} - \alpha^{k-1}}{\beta}\]. Where \(\displaystyle{\alpha}\) and \(\displaystyle{\beta}\) are positive coprime integers. Give their sum.

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