2010/033) Let S be the sum of all the real coefficients of the expansion of (1 + ix)^2009. what is log_2(S)?

Let S be the sum of all the real coefficients of the expansion of (1 + ix)^2009. what is log_2(S)?

the sum of all the real coefficients of the expansion of (1 + ix)^2009 is real part of (1+i)^2009 by putting x = 1

now (1+i) = sqrt(2) cis(pi/4) where cis(x) = cos x + i sin x

so (1+i)^2009 = sqrt(2)^2009(cis (2009pi/4))

so real part = sqrt(2)^2009 cos(2009pi/4) = sqrt(2)^2009 * cos(pi/4) = sqrt(2)^2008 = 2^1004

hence log_2(s)= 1004