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Question

If ω is a non-real cube root of unity, then the value of 1(2ω)(2ω2)+2(3ω)(3ω2)+ 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 B 44080
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+316(n+1)n(2n+1)+312(n+1)n
=14(n+1)n(n2+5n+8)
Given n=19
S=141920464=44080

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