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Question

6k=1(sin2πk7icos2πk7) equals

A
1
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B
i
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C
-i
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D
-1
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Solution

The correct option is B i
Solution:
6k=1(sin2πk7icos2πk7)
=6k=1i(cos2πk7+isin2πk7)
=i6k=1ei2kπ7
=iei2π7+ei4π7+ei6π7+ei8π7+ei10π7+ei12π7
=i⎢ ⎢ ⎢ei2π7⎜ ⎜ ⎜1ei12π71ei2π7⎟ ⎟ ⎟⎥ ⎥ ⎥

=i⎢ ⎢ ⎢⎜ ⎜ ⎜ei2π7ei14π71ei2π7⎟ ⎟ ⎟⎥ ⎥ ⎥

=i⎢ ⎢ ⎢⎜ ⎜ ⎜ei2π711ei2π7⎟ ⎟ ⎟⎥ ⎥ ⎥ (ei14π7=1)

=i×(1)
=i
Hence, A is the correct option.

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