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If omega is a cube root of unity then th...

If `omega` is a cube root of unity then the value of the expression `2(1+w)(1+w^2)+3(2+w)(2+w^2)+.....+(n+1)(n+w)(n+w^2)`

Text Solution

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We can write the given expression as ,
`S = sum_(m=1)^n(m+1)(m+omega)(m+omega^2)`
So , general term in this expression can be given as,
`T_m = (m+1)(m+omega)(m+omega^2)`
`= (m+1)(m^2+(omega+omega^2)m+omega^3)`
As, `1+omega+omega^2 = 0=>omega+omega^2 = -1`
`:. T_m = (m+1)(m^2-m+1)` (As `omega^3 = 1`)
`=> T_m = (m+1)(m(m-1)+1)`
...
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