The function for 2016

Algebra Level 5

For all positive integers \(x\), a function \(f(x)\) is defined as follows:

  • \(f(1)=2016\).
  • \(f(1)+f(2)+f(3)+\cdots + f(n-1)+f(n)={ n }^{ 2 }f(n)\) for \(x>1\).

If the value of \(f(2016)\) can be expressed as \(\dfrac pq\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

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