Radical radicals

Algebra Level 5

The sequence \(\{u_n\}\subset\mathbb{Z}^+\) on \(n\in\mathbb{Z}^+\) is defined by \(u_1=1\) and

\[\sqrt{u_n+\sqrt{u_{n-1}+\sqrt{u_{n-2}+\sqrt{...+\sqrt{u_2+\sqrt{u_1}}}}}}=n\]

The sequence \(\{v_n\}\subset\mathbb{Z}^+\) on \(n\in\mathbb{Z}^+\) is defined by \(v_1=1\) and

\[\sqrt[3]{v_n+\sqrt[3]{v_{n-1}+\sqrt[3]{v_{n-2}+\sqrt[3]{...+\sqrt[3]{v_2+\sqrt[3]{v_1}}}}}}=n\]

Let:

  • \(A\) = the last digit of \(u_{2016}\)
  • \(B\) = \(\text{min}(k)\) such that \(u_k-k \geq 2016\)
  • \(C\) = \(\text{min}(k)\) such that \(v_k-u_k \geq 2016\)

Find \(A+B+C\).

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