Solution: obtained by A4+A3+A2+A+ 1 = 0.
(a5- 1)/(a- 1)=0
That is A5 = 1, a≠ 1.
a2005+a2006+ 1 =(a5)40 1+a(a5)40 1+ 1
=a+2
There are four answers: (all imaginary numbers)
2+cos72 +isin72
2+cos 144+is 144
2+cos2 16 +isin2 16
2+cos288 +isin288