Question

If $1, \alpha_{1}, \alpha_{2}, \ldots \ldots, \alpha_{2008}$ are (2009 roots of unity, then the value of $\Sigma_{ r =1}^{2008} r \left(\alpha_{ r }+\alpha_{2009- r }\right)$ equals:

• A. $2009$
• B. $2008$
• C. $0$
• D. $-2009$

Answer 1

Answer: (D)

$\displaystyle \sum _{r = 1}^{2008}r\left[\alpha _{r} + \alpha _{2009 - r}\right]=1\left[\alpha _{1} + \alpha _{2008}\right]+2\left[\alpha _{2} + \alpha _{2007}\right]+\ldots \ldots \ldots +2008\left[\alpha _{2008} + \alpha _{1}\right]$
$=2009\left[\alpha _{1} + \alpha _{2} + \alpha _{3} + \ldots \ldots \alpha _{2008}\right]=-2009$