jmc

Let $$\begin{array}{rl}S& =2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots +\frac{1}{2}\left(3+\frac{1}{2}\cdot 2\right)\right)\cdots \right)\\ & =2002+\frac{2001}{2}+\frac{2000}{2^2}+\cdots +\frac{3}{2^{1999}}+\frac{2}{2^{2000}}.\end{array}$$Then $$2S=2\cdot 2002+2001+\frac{2000}{2}+\cdots +\frac{3}{2^{1998}}+\frac{2}{2^{1999}}.$$Subtracting these equations, we get $$\begin{array}{rl}S& =4004-1-\frac{1}{2}-\frac{1}{2^2}-\cdots -\frac{1}{2^{1999}}-\frac{2}{2^{2000}}\\ & =4004-1-\frac{1}{2}-\frac{1}{2^2}-\cdots -\frac{1}{2^{1999}}-\frac{1}{2^{1999}}\\ & =4004-\frac{1}{2^{1999}}(2^{1999}+2^{1998}+\cdots +2+1+1)\\ & =4004-\frac{1}{2^{1999}}\cdot 2^{2000}\\ & =4004-2=\boxed{4002}.\end{array}$$