G2.1 If $R=1 \times 2+2 \times 2^{2}+3 \times 2^{3}+\ldots+10 \times 2^{10}$, find the value of $R$. Similar question: 2005 HI7 If $M=\sum_{n=1}^{1024}\left[\log _{2} n\right]$, find the value of $M$.
$\begin{array}{l}2 R=\quad 1 \times 2^{2}+2 \times 2^{3}+\ldots+9 \times 2^{10}+10 \times 2^{11} \\ R-2 R=2+2^{2}+2^{3}+\ldots \ldots+2^{10}-10 \times 2^{11} \\ -R=\frac{a\left(R^{n}-1\right)}{R-1}-10 \times 2^{11}=\frac{2\left(2^{10}-1\right)}{2-1}-10 \times 2048 \\ R=20480-2(1023)=18434\end{array}$