2. Determine the sum of the numbers

$$
1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+5 \cdot 2^{4}+\ldots+2001 \cdot 2^{2000}
$$
2. Let's denote this sum by S. Then we have

$$
\begin{array}{r}
\mathrm{S}=1+2+2^{2}+2^{3}+\ldots+2^{1999}+2^{2000} \\
+2+2^{2}+2^{3}+\ldots+2^{1999}+2^{2000} \\
+2^{2}+2^{3}+\ldots+2^{1999}+2^{2000} \\
+2^{3}+\ldots+2^{1999}+2^{2000} \\
\vdots+2^{1999}+2^{2000} \\
+2^{2000}
\end{array}
$$

$$
\begin{aligned}
\mathrm{S} & =\frac{1 \cdot\left(2^{2001}-1\right)}{2-1}+\frac{2 \cdot\left(2^{2000}-1\right)}{2-1}+\ldots+\frac{2^{1999} \cdot\left(2^{2}-1\right)}{2-1}+2^{2000}(2-1) \\
& =2001 \cdot 2^{2001}-\left(1+2+2^{2}+\ldots+2^{1999}+2^{2000}\right) \\
& =2001 \cdot 2^{2001}-\frac{2^{2001}-1}{2-1} \\
& =2000 \cdot 2^{2001}+1
\end{aligned}
$$

20 points