If α≠1 is any of 7th roots of unity then real part of α2009 + 3α2010 + 5α2011 + ------ + up to 7terms is

Let S=α2009+3α2010+5α2011+−−−−−+13α2015 =1+3α+5α2+−−−−−−−+13α6(AGP)=−141−α=−141−cis2Kπ7=−71+icot⁡Kπ7

If α≠1 is any of 7th roots of unity then real part of α2009 + 3α2010 + 5α2011 + ------ + up to 7terms is

Let S=α2009+3α2010+5α2011+−−−−−+13α2015 =1+3α+5α2+−−−−−−−+13α6(AGP)=−141−α=−141−cis2Kπ7=−71+icot⁡Kπ7

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If $α \neq$ 1 is any of 7^th roots of unity then real part of $α$ ²⁰⁰⁹ + 3 $α$ ²⁰¹⁰ + 5 $α$ ²⁰¹¹+ ------ + up to 7
terms is

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-14

answer is C.

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Detailed Solution

$Let S = α^{2009} + 3 α^{2010} + 5 α^{2011} + - - - - - + 13 α^{2015} \begin{array}{l} = 1 + 3 α + 5 α^{2} + - - - - - - - + 13 α^{6} (AGP) \\ = \frac{- 14}{1 - α} = \frac{- 14}{1 - cis \frac{2 Kπ}{7}} \\ = - 7 [1 + icot \frac{Kπ}{7}] \end{array}$