If represents a complex cube root of unity, then (1+ 1 ) (1+ 1 ^2 )+ (2+ 1 ) (2+ 1 ^2 )+ + (n+ 1 ) (n+ 1 ^2 )=

Question

Accepted Answer

Correct answer: B. (b) Given that is a complex cube root of unity ∴1+ + ^2=0 and ^3=1 ∵(k+ 1 )(k+ 1 ^2)=k^2+k( 1 + 1 ^2)+ 1 ^3 =k^2+ ( ^2+ ^3 )k+ 1 ^3=k^2-k+1 … . (i ) [From Eq. (i)] Now, _k=1^n 0.05em(k+ 1 )(k+ 1 ^2) = 0.05em 0.05em(k^2-k+1)= n (n+1 ) (2n+1 )6- n (n+1 )2+n = n (n+1 )6 [2n+1-3 ]+n= n (n+1 )6 (2n-2 )+n = n3(n^2-1)+n= n3(n^2-1+3)= n (n^2+2 )3

$If ω represents a complex cube root of unity, then (1 + \frac{1}{ω}) (1 + \frac{1}{ω ^{2}}) + (2 + \frac{1}{ω}) (2 + \frac{1}{ω ^{2}}) + \dots + (n + \frac{1}{ω}) (n + \frac{1}{ω ^{2}}) =$

Solution

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