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  • If omega is a complex cube root of unity , then value of expression cos[ (1-omega)(1-omega^2) +............+(10-omega)(10-omega^2) pi/900]

If omega is a complex cube root of unity , then value of expression cos[ (1-omega)(1-omega^2) +............+(10-omega)(10-omega^2) pi/900]

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Solution: We have

         (r-omega )(r-omega ^2)=r^2-(omega +omega ^2)r+1\ \Rightarrow hspace0.2cm=r^2+r+1=(r+1)^2-r

          herefore     sum_r=1^10(r-omega )(r-omega ^2)=sum_r=1^10[(r+1)^2-r]

          \Rightarrow (1/6)(11)(11+1)(22+1)-1-(1/2)(10)(11)=450

 

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Posted by

Deependra Verma

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