Inspired by Calvin Lin - Part I

Algebra Level 5

$\large f_n(x)=x^2 + ( x + 1) ^2 +(x+2)^2+\ldots+(x+n)^2$

The function $$f_n(x)$$ is defined above for $$x$$ ranges over all real values with natural number $$n$$.

If the minimum value of $$f_n(x)$$ is $$(2^{10}-12)$$ then find the value of $$n$$.

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