If ${\left(3+{\mathrm{x}}^{2008}+{\mathrm{x}}^{2009}\right)}^{2010}={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{x}+{\mathrm{a}}_2{\mathrm{x}}^2+\cdots +{\mathrm{a}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}$, then the value of ${\mathrm{a}}_0-\frac{1}{2}{\mathrm{a}}_1-\frac{1}{2}{\mathrm{a}}_2+{\mathrm{a}}_3-\frac{1}{2}{\mathrm{a}}_4-\frac{1}{2}{\mathrm{a}}_5+{\mathrm{a}}_6-\dots$ is

Put $\mathrm{x}=\mathrm{\omega },{\mathrm{\omega }}^2$

$\begin{array}{c}{\left(3+\mathrm{\omega }+{\mathrm{\omega }}^2\right)}^{2010}={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{\omega }+{\mathrm{a}}_2{\mathrm{\omega }}^2+\cdots \\ \mathrm{and} \left(3+{\mathrm{\omega }}^2+{\mathrm{\omega }}^4\right)={\mathrm{a}}_0+{\mathrm{a}}_1{\mathrm{\omega }}^2+{\mathrm{a}}_2{\mathrm{\omega }}^4+\dots \\ \Rightarrow 2^{2010}={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{\omega }+{\mathrm{a}}_2{\mathrm{\omega }}^2+{\mathrm{a}}_3+{\mathrm{a}}_4\mathrm{\omega }+\cdots \left(1\right)\\ \mathrm{and} 2^{2010}={\mathrm{a}}_0+{\mathrm{a}}_1{\mathrm{\omega }}^2+{\mathrm{a}}_2\mathrm{\omega }+{\mathrm{a}}_3+{\mathrm{a}}_4{\mathrm{\omega }}^2+\cdots \left(2\right)\end{array}$

Adding (1) and (2), we have

$\begin{array}{c}2\times 2^{2010}=2{\mathrm{a}}_0-{\mathrm{a}}_1-{\mathrm{a}}_2+2{\mathrm{a}}_3-{\mathrm{a}}_4-{\mathrm{a}}_5+2{\mathrm{a}}_6-\cdots \\ \mathrm{or} 2^{2010}={\mathrm{a}}_0-\frac{1}{2}{\mathrm{a}}_1-\frac{1}{2}{\mathrm{a}}_2+{\mathrm{a}}_3-\frac{1}{2}{\mathrm{a}}_4-\frac{1}{2}{\mathrm{a}}_5+{\mathrm{a}}_6\cdots \end{array}$