If 1,ωandω2are the cube roots of unity, then 1-ω+ω21-ω2+ω41-ω4+ω81-ω8+ω16......upto2n factors is
2n
22n
1
-22n
Explanation of the correct option:
Step1. Define the cube root of unity:
Given,
1,ωandω2are the cube root of unity.
factor1+ω+ω2=0____(1)Fromdefinitionω=-1+i32,ω2=-1-i321×ω×ω2=1_____(2)
Step2. Find the value of each term (1-ω+ω2)(1-ω2+ω4)(1-ω4+ω8)(1-ω8+ω16)......upto2n factor:
1-ω+ω2=1+ω2-ω=-ω-ωFrom(1),1+ω2=-ω=-2ω1-ω2+ω4=1-ω2+ω3×ω=1-ω2+1×ω[From(2),(ω3=1)]=1-ω2+ω=-ω2-ω2[From(1)]=-2ω21-ω4+ω8=1-ω3ω+ω32ω2=1-ω+ω2[From(2),ω3=1]=-ω-ω[From(1),1+ω2=-ω]=-2ω1-ω8+ω16=1-ω32ω2+ω35ω=1-ω2+ω=-ω2-ω2=-2ω2.......soonupto2nfactors
Step3. Find the value of (1-ω+ω2)(1-ω2+ω4)(1-ω4+ω8)(1-ω8+ω16)......upto2nfactors:
(1-ω+ω2)(1-ω2+ω4)(1-ω4+ω8)(1-ω8+ω16)......upto2nfactors=-2ωn-2ω2n[Substitutethevalueofrequiredterms]=4ωω2n=4ω3n∵ω3=1=4n=22n
Hence, Option (B) is the correct answer.