If x2+x+1=0, then the value of x+1x2+x2+1x22+x3+1x32+⋯+x27+1x272 is

x2+x+1=0⇒ x=ω or ω2Let x = ω. Then,x+1x=ω+1ω=ω+ω2=−1x2+1x2=ω2+1ω2=ω2+ω=−1x3+1x3=ω3+1ω3=2x4+1x4=ω4+1ω4=ω+ω2=−1, etc. ∴ x+1x2+x2+1x22+x3+1x32+⋯+x27+1x272=x+1x2+x2+1x22+x4+1x42+⋯+x26+1x262+x3+1x32+x6+1x62+x9+1x92+⋯+x27+1x272 =18+922=54