# Radical radicals

Algebra Level 4

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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