# Math Help - Quadratic Equation

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

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

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 = $\frac{a(1-r^{n})}{1-r}$ where r is common ratio .

The equation $\alpha^{2}+\alpha+1=0$ has complex roots, so can't you simply write (them) in exponential form and substitute into $\alpha^{2010}$ ?