To solve this problem, you need to learn a formula: a5-1= (a-1) (a4+a3+a2+a+1).

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