\(nx^2-x+n(n+1)=0\)

Algebra Level 3

For all positive integers \(n\), let \(\alpha_n\) and \(\beta_n\) be the roots of the quadratic equation \[nx^2-x+n(n+1)=0.\] What is the value of \[\left(\frac{1}{\alpha_1}+\frac{1}{\beta_1}\right)+\left(\frac{1}{\alpha_2}+\frac{1}{\beta_2}\right)+ \cdots +\left(\frac{1}{\alpha_{25}}+\frac{1}{\beta_{25}}\right)?\]

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