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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)` |
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Answer» 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)` `=m(m^2-1)+m+1` `:. T_m=m^3+1` `:. S = sum_(m=1)^n (m^3+1) = sum_(m=1)^n m^3+ sum_(m=1)^n 1` `=> S= ((n(n+1))/2)^2+n = 1/4(n^2)(n+1)^2+n` |
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