A year problem by Nihar

Algebra Level 4

\[\large \left(\dfrac{x(x+1)}{2}\right)^2 - \left(\dfrac{x(x-1)}{2}\right)^2=\dfrac{x^2}{n}\]

Let \(S_{n}\) be the sum of all values of \(x\) that satisfy the above equation where \(n \ne 0 \).

If \(S_{2014} \) can be expressed as \(\dfrac{a}{b}\) where \(a,b\)are coprime integers, then what is the value of \(a+b\)?

Bonus: generalize this for all \(n\).

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