If $\displaystyle \alpha$ is root of equation $\displaystyle x^2+x+1 = 0$ then find the value of $\displaystyle 1+\alpha +\alpha^2+\alpha^3+.....+\alpha^{2010}$

Here I have put the value of $\displaystyle \alpha$ in the given equation to get $\displaystyle 1+\alpha + \alpha^2$ which is similar to the first three terms. So, each three terms give value = 0 . Only the last term will remain which is $\displaystyle \alpha^{2010}$

Can we equate this with the help of Geometric progression somehow....as the given terms form a G.P with first term 1 and common ratio

Sum of the n terms of G.P = $\displaystyle \frac{a(1-r^{n})}{1-r}$ where r is common ratio .

Please suggest.