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Suppose that w is the imaginary ${(2009)}^{t h}$ roots of unity . if $(2^{2009} - 1) \sum_{r = 1}^{2008} \frac{1}{2 - w^{r}} = (a) (2^{b}) + c w h e r e a, b, c \in$ N, and the least value of (a+b+c) is (2008)k.The numerical value of K is

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answer is B.

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Detailed Solution

$∴ (2^{2009} - 1) \sum_{r = 1}^{2008} \frac{1}{2 - w^{r}} = {2009.2}^{2008} - 2^{2009} + 1$
$= 2^{2008} (2009 - 2) + 1 = 2^{2008} .2007 + 1 = [(a) (2^{b}) + c]$
$a = 2007, b = 2008, c = 1$
A+b+c=4016

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