Find all the possible values for: $1+a+a^2+\cdots+a^{2011}$

Let $x=a$ be a solution of the equation $x^{2012}-7x+6=0$. Find all the possible values for: $1+a+a^2+\cdots+a^{2011}$.

We're told $x=a$ is a solution of the equation $x^{2012}-7x+6=0$, therefore we have $a^{2012}-7a+6=0$.

It can be rewritten as

$a^{2012}-1-7a+6+1=0$

$a^{2012}-1-7a+7=0$

$(a^{2012}-1)-7(a-1)=0$