RIOPLATENSE 2001 Problem 3

For every integer $n > 1$ , the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor$ where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$ . Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$ . .