The value of the expression `( 1+ 1/w) (1+ 1/w^2) + ( 2+ 1/w) (2 + 1/w^2) + ( 3+ 1/w) (3+ 1/w^2) +........+( n+ 1/w) (n+ 1/w^2)`, where w is an imaginary cube root of unity

Explanation

`(1+1/omega)(1+1/omega^2) = 1+1/omega^2+1/omega+1/omega^3`

`=(omega^3+omega+omega^2+1)/omega^3`

As, `omega^3 = 1 and 1+omega+omega^2 = 0`

`:. (1+1/omega)(1+1/omega^2) = 1`

`(2+1/omega)(2+1/omega^2) = 4+2/omega^2+2/omega+1/omega^3`

`=(4omega^3+2omega+2omega^2+1)/omega^3`

As, `omega^3 = 1 and 1+omega+omega^2 = 0`

`(2+1/omega)(2+1/omega^2) = 2+1 = 3`

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