Inspired by Calvin Lin - Part I

Algebra Level 5

fn(x)=x2+(x+1)2+(x+2)2++(x+n)2 \large f_n(x)=x^2 + ( x + 1) ^2 +(x+2)^2+\ldots+(x+n)^2

The function fn(x)f_n(x) is defined above for xx ranges over all real values with natural number nn.

If the minimum value of fn(x)f_n(x) is (21012)(2^{10}-12) then find the value of nn.

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