Trignometry+Summation

$\displaystyle \sum_{n=1}^{50} \tan^{-1} \left(\dfrac{n + n}{1 + n^4 - n^2}\right)$

$\tan^{-1}a - tan^{-1} b = tan^{-1} \left(\dfrac{a - b}{1 + ab}\right)$

$\displaystyle \sum_{n=1}^{50} \tan^{-1} \left(\dfrac{n^2 + n - (n^2 - n)}{1 + n^4 - n^2}\right)$

$\displaystyle \sum_{n=1}^{50} \tan^{-1} (n^2 + n) - \tan^{-1} (n^2 - n)$

$= \tan^{-1} 2 - \tan^{-1} 0 + \tan^{-1} 6 - \tan^{-1} 2 + \cdots + \tan^{-1} 2550 - \tan^{-1} 2450$

$= \tan^{-1} 2550$