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Let S be the sum of all the real coefficients of the expansion of \({(1+ix)}^{2009}\). What is \(\log_{2}(S)\)?

 Jan 25, 2019
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\(\text{The sum of all the real coefficients is simply the real part of }\\ \left . (1+ix)^{2009} \right |_{x=1}\)

 

\((1+i)^{2009} = \sqrt{2}^{2009}e^{i 2009 \left(\frac \pi 4\right)} = \sqrt{2}^{2009}\left(\cos\left(\dfrac \pi 4\right)+i \sin\left(\dfrac \pi 4 \right)\right)\)

 

\(Re\left[(1+i)^{2009}\right] = \sqrt{2}^{2009} \cdot \dfrac{\sqrt{2}}{2} = 2^{1004}\\ \log_2\left(2^{1004}\right) = 1004\)

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 Jan 25, 2019
edited by Rom  Jan 25, 2019

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