IMC 1995 Problem 8

Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$ , $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$ . Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$