JEE Advance - Mathematics (1996 - No. 12)
The value of the expression
$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right),$$
$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right),$$
where $$\omega $$ is an imaginary cube root of unity, is..........
$$\frac{1}{4}n(n-1)(n^2 + 3n + 4)$$
$$\frac{1}{6}n(n+1)(n^2 + n + 1)$$
$$\frac{1}{3}n(n-1)(n^2 + 2n + 3)$$
$$\frac{1}{2}n(n+1)(n^2 + 3n + 2)$$
$$\frac{1}{5}n(n-1)(n^2 + 4n + 5)$$
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