If ω is a non-real cube root of unity, then the value of 1⋅(2−ω)(2−ω2)+2⋅(3−ω)(3−ω2)+⋯ sum upto 19 terms is
A
56680
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B
44080
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C
84390
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D
64580
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Solution
The correct option is B44080 rth term of the given series is =r[(r+1)−ω][(r+1)−ω2] =r[(r+1)2−(ω+ω2)(r+1)+ω3] =r[(r+1)2−(−1)(r+1)+1] =r[r2+3r+3]=r3+3r2+3r
Thus, sum of the given series S=(n)∑r=1(r3+3r2+3r) =14(n+1)2n2+3⋅16(n+1)n(2n+1)+3⋅12(n+1)n =14(n+1)n(n2+5n+8)
Given n=19 ∴S=14⋅19⋅20⋅464=44080